Class 8th Mathematics AP Board Solution
Exercise 1.1- Name the property involved in the following examples
- 2 (3/5 + 1/2) = 2 (3/5) Name the property involved in the following examples…
- 3/7 x 1 = 3/7 = 1 x 3/7 Name the property involved in the following examples…
- (-2/5) x 1 = -2/5 Name the property involved in the following examples…
- 2/5 + 1/3 = 1/3 + 2/5 Name the property involved in the following examples…
- 5/2 x 3/7 = 15/14 Name the property involved in the following examples…
- 7a + (-7a) = 0 Name the property involved in the following examples…
- x x 1/x = 1 (x not equal 0) Name the property involved in the following…
- (2 x x) + (2 x 6) = 2 x (x+6) Name the property involved in the following…
- -3/5 Write the additive and the multiplicative inverses of the following.…
- 1 Write the additive and the multiplicative inverses of the following.…
- 0 Write the additive and the multiplicative inverses of the following.…
- 7/9 Write the additive and the multiplicative inverses of the following.…
- −1 Write the additive and the multiplicative inverses of the following.…
- Fill in the blanks.
- Fill in the blanks.
- 1 x 9/11 Fill in the blanks.
- Fill in the blanks.
- Multiply by the reciprocal of -5/14
- Which properties can be used in computing 2/5 x (5 x 7/6) + 1/3 x (3 x 4/11)…
- Verify the following (5/4 + -1/2) + -3/2 = -5/4 + (-1/2 + -3/2)
- Evaluate after rearrangement.
- 3/4 from 1/3 Subtract
- -32/13 from 2 Subtract
- −7 from -4/7 Subtract
- What numbers should be added to -5/8 so as to get -3/2 .
- The sum of two rational numbers is 8 If one of the numbers is -5/6 find the…
- Is subtraction associative in rational numbers? Explain with an example.…
- x = 2/15 Verify that - (-x) = x for
- x = -13/17 Verify that - (-x) = x for
- Write- (i) The set of numbers which do not have any additive identity (ii) The…
Exercise 1.2- Represent these numbers on the number line. (i) 9/7 (ii) -7/5
- Write five rational numbers which are smaller than 5/6 .
- Find 12 rational numbers between -1 and 2.
- Find a rational number between 2/3 and 3/4 . [Hint: First write the rational…
- Find ten rational numbers between - 3/4 and 5/6 .
Exercise 1.3- Express each of the following decimal in the p/q form. (i) 0.57 (ii) 0.176 (iii)…
- Express each of the following decimals in the rational form (p/q) .…
- Express each of the following decimals in the rational form (p/q) . 0. bar 57…
- Find (x + y) (x y) if
- Find (x + y) ÷ (x − y) if x = 1/4 , y = 3/2
- Divide the sum of - 13/5 and 12/7 by the product of - 13/7 and - 1/2 .…
- If of a number exceeds 1/7 of the same number by 36. Find the number.…
- Two pieces of lengths 2 3/5 m and 3 3/10 m are cut off from a rope 11 m long.…
- The cost of meters of cloth is 12 3/4 . Find the cost per metre.
- Find the area of a rectangular park which is 18 3/5 m long and 8 2/3 m broad.…
- What number should be divided by to get - 11/4 ?
- If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is…
- When the repeating decimal 0.363636 .... is written in simplest fractional form…
- Name the property involved in the following examples
- 2 (3/5 + 1/2) = 2 (3/5) Name the property involved in the following examples…
- 3/7 x 1 = 3/7 = 1 x 3/7 Name the property involved in the following examples…
- (-2/5) x 1 = -2/5 Name the property involved in the following examples…
- 2/5 + 1/3 = 1/3 + 2/5 Name the property involved in the following examples…
- 5/2 x 3/7 = 15/14 Name the property involved in the following examples…
- 7a + (-7a) = 0 Name the property involved in the following examples…
- x x 1/x = 1 (x not equal 0) Name the property involved in the following…
- (2 x x) + (2 x 6) = 2 x (x+6) Name the property involved in the following…
- -3/5 Write the additive and the multiplicative inverses of the following.…
- 1 Write the additive and the multiplicative inverses of the following.…
- 0 Write the additive and the multiplicative inverses of the following.…
- 7/9 Write the additive and the multiplicative inverses of the following.…
- −1 Write the additive and the multiplicative inverses of the following.…
- Fill in the blanks.
- Fill in the blanks.
- 1 x 9/11 Fill in the blanks.
- Fill in the blanks.
- Multiply by the reciprocal of -5/14
- Which properties can be used in computing 2/5 x (5 x 7/6) + 1/3 x (3 x 4/11)…
- Verify the following (5/4 + -1/2) + -3/2 = -5/4 + (-1/2 + -3/2)
- Evaluate after rearrangement.
- 3/4 from 1/3 Subtract
- -32/13 from 2 Subtract
- −7 from -4/7 Subtract
- What numbers should be added to -5/8 so as to get -3/2 .
- The sum of two rational numbers is 8 If one of the numbers is -5/6 find the…
- Is subtraction associative in rational numbers? Explain with an example.…
- x = 2/15 Verify that - (-x) = x for
- x = -13/17 Verify that - (-x) = x for
- Write- (i) The set of numbers which do not have any additive identity (ii) The…
- Represent these numbers on the number line. (i) 9/7 (ii) -7/5
- Write five rational numbers which are smaller than 5/6 .
- Find 12 rational numbers between -1 and 2.
- Find a rational number between 2/3 and 3/4 . [Hint: First write the rational…
- Find ten rational numbers between - 3/4 and 5/6 .
- Express each of the following decimal in the p/q form. (i) 0.57 (ii) 0.176 (iii)…
- Express each of the following decimals in the rational form (p/q) .…
- Express each of the following decimals in the rational form (p/q) . 0. bar 57…
- Find (x + y) (x y) if
- Find (x + y) ÷ (x − y) if x = 1/4 , y = 3/2
- Divide the sum of - 13/5 and 12/7 by the product of - 13/7 and - 1/2 .…
- If of a number exceeds 1/7 of the same number by 36. Find the number.…
- Two pieces of lengths 2 3/5 m and 3 3/10 m are cut off from a rope 11 m long.…
- The cost of meters of cloth is 12 3/4 . Find the cost per metre.
- Find the area of a rectangular park which is 18 3/5 m long and 8 2/3 m broad.…
- What number should be divided by to get - 11/4 ?
- If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is…
- When the repeating decimal 0.363636 .... is written in simplest fractional form…
Exercise 1.1
Question 1.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer: Role of zero/additive identity:
0 is the additive identity for rational numbers, i.e. if 'a' is any rational number then
a + 0 = a = 0 + a
Question 2.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:Distributive law
This property is know as the distributive law of multiplication over addition.
For all rational numbers a, b and c
a(b + c) = ab + ac
⇒![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAOcAAAApCAYAAADOFj7kAAAFeElEQVR4nO2cTYgcRRTHf5eBERYCiyAoCRhR2CUuxA0JOCs0Kiyag4HNQUGFFQRzMDcRQXBAXUGRBFQCQvxCD4ofhxgUkYgiRFHxIJ70kMN6yMGLh1w8jIc37fTW9Ef1dHfV6+76QZFM70z3q/f/d1NV/bohEAgEAoGADobAMeA4cNBzLCZrSHyBbDTr55LOeWUN2AGOACvAZeADYOAxpv3AJvAq8C9wq8dYtKNRP5d02itfAF8nPm8CE+Ckn3AASfg6cP80lk4lvGY06ueSTnvlFeAbZlfaWNzHvUU0Y0QHE14zmvVzSS+8sgNcBW7wHQg9SXjNaNLPJZ31ygA4ADwEfI8sLmigswmvGa36uaSzXhkinXsQuAQ86jec/+lswmtGq34u6YVXIqSTj3mOA3qS8JqJ0KOfS1rrlS3g3oy/rQPLic9DpJO/NR2UBTYJP1Xw96bIy6lLNOvnEnVeiYcz95C9ALANvE76fa91pEMXjO0T4M+aYqyCTcKXkNXKVScRCWk5tdGibjTp56P/SVR5ZRV4G0nGSeAv4CXjO0eBH8iumjgAXAFOJLYdQTr5YoXYTlX4bRLbocoq8DtuqkPScmqjRRNo0c9X/5Oo8sol4wB3IcElE/sjcF/BfraBs8ChabsMvEO14CfAdRV+vwaMgW+n+7o4/Xxzzm/eoJohbUnLqY0WTaFBP5/9V+eVm6aBfGhs/wf4avr/CLmC2ZRxLQMPAA9Tz5i86sm5CLcDfyNDl6aImM+pjRYm75E9tBog5lkpEZdP/TT0vyyNemUInEeumkmuMUvIOeCtJg5ugY+TE+TEOd7g/tNyaqOFyQYyJzQNOgA+As7gtza2jH5t7X/TXtlDxN6yrV+B064ObuDr5PwMeC1l+yPIUMe2mUaLsc1pRHEJnWlQLScmVNcvQn//s7xSOwNkzP0ps45dQ+osk4wrNlvKiFtnTG8CH6ccYwVZMLBthzJiTcupSZoWWcQGPQp8R7ExxxWbLVVOzrb0P8srtfMckozkpHwC3G18b1yxmTyFDGnMNgHeTdn+TMo+6oxpjIjcFGk5NUnTIo+IWb6KzDyu2Ezq0M+kLf0f06xXABmCnWE+GTZGqsr1yHK+2SbALSnbm77/dY7seU4dFOU0S4ss4qHc58gcyOW9Wqhfvzb1v2mvcAJ4nr1XnLPTf3eRuZYPfM053yd9EWwHEcK2vZyx/7yc5mmRhjnH2sLPCZrGIvq1rf9ZXqmFDWQoEM+TNpEJeJyQC8ALTR28AF8n5080uwiWldMiLUyyFj+0nKBl9Wtj/xvzyj6kE5OUFl/ZTyOlSj7wcXIuIa+paPIdOmk5tdHCJG9VMjbo/hriXZQy+rWx/y68kssysrp4Y8H39jEf5GH8VghB+VrNbeDLiscswjanRYzIX/w4jOhig1b98qiz/6DTK4U8i1Rb5BHXJF5ByrB2yS6Ut2WXauKWrdUcAL8Ad1Y4pi02OXWJRv1cotkruQyQWsu8J+NHyEuiziM3ZTccxFVE2VrNp3E3v7bJqUs06ucSzV4pZAn4hOxJ9ghdL4MqW6u5hVwpXVbVFOXUJdr0c0kbvFJI/I6ZNGJxh9Pv+H4pb9lazbynD5okL6cu0aafS9rilYUZIZPjJ5Fx+0X0vfM0op+ve7ShDfq5JKJDXjnI3tdt3IZ07g4/4cxRplazj2jXzyW98Mou8sydBsrWagZ06eeSTnlliNwMNp/s/wMHRcEWlK3V7Bva9XNJ57wSFzmbtaRXmV8Bc03ZWs0+olk/l3TWK2bx7ypS2tT0Ey15lK3V7DMa9XNJp71yDKkoWUHe3PYz8ITHeBap1ewz2vRzSfBKIBAIBAKBQCAQCAQCgUAgEAgEEvwHdxIME/4pgdQAAAAASUVORK5CYII=)
Question 3.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:Multiplicative identity
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.
Question 4.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:Multiplicative identity
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.
Question 5.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:Commutative law of addition
In commutative law of addition, a, b is rational number where
a + b = b + a
Question 6.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:Closure law in multiplication
We find that rational numbers are closed under multiplication. For any two rational numbers a and b, a
b is also rational number.
Question 7.Name the property involved in the following examples
7a + (-7a) = 0
Answer:(vii) Additive inverse law
Any two numbers whose sum is 0 are called the additive inverses of each other. In general if ‘a’ represents any rational number then a + (-a) = 0 and (-a ) + a = 0
Then a, (-a) are additive inverse of each other.
Question 8.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:(viii) Multiplicative inverse
We say that a rational number
is called the reciprocal or the multiplicative inverse of another rational number
if ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFYAAAAgCAMAAAC/838ZAAAAAXNSR0IArs4c6QAAAIdQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGaQAGa2OgAAOgA6OgBmOjo6OmaQOma2OpDbZgAAZjoAZjo6ZjpmZmaQZpDbZrbbZrb/kDoAkGYAkGY6kNv/tmYAtmY6tmaQtpA6tpBmttv/tv//25A627Zm27aQ29u22////7Zm/9uQ/9u2//+2///bYo78OgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABTUlEQVRIS+1V21LCMBDNgi1FRVu8RSgiRklz+f/vc9ukTEHSTZjxjX3JTHN6dvfsJYxdDRXQK4BHUgmLqPyLhB0Apnxlm1wSP1heSHWXQNvyNSStul3HR9ohv+8xPSraZNpmsmYio9IzZSHtB+V8kJDhkH1CQaWoKsiidPhZRsEof8f3uoLp+czNk/uu/ZnCqxY7EaC1vFPalA8phAdsiJZZjo2BNbqIlQVpW959VYyWvYbesJGOzNP218M7y086+RwokEw4WgyXHpCQRIQI44NHivDXa1+yhJkakIQbrAv00lYQpzX0LvWzC1P5M6XP7GqO1b15YUzQ+8ARRwMdvI7t+2hgGq2I9e9oZxuYvBEK4maCWLW8CNPdyCA7TPuU2fe0aBFNvSndfaII/0SbS7sN7N5ecVy+UvNZytzV+PKSz1Qzh8UWsM2vFq/AL3mPF//e7j77AAAAAElFTkSuQmCC)
Question 9.Name the property involved in the following examples
![](data:image/png;base64,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)
Answer:Distributive property
This property is known as distribution law of multiplication over addition. For all rational numbers a, b and c
a + b = b + a
Question 10.Write the additive and the multiplicative inverses of the following.
![](data:image/png;base64,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)
Answer:Additive inverse, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAiCAMAAACk7TNkAAAAAXNSR0IArs4c6QAAAFRQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgA6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ/9uQ/9u2//+2///bXfyARAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAY0lEQVQoU42PSw6AIBBDC+L/gygoyv3v6TAY3WjC2zSZtGkH+MNVQnZAmGZs0rDLl6yH7qMsomAFdjXgbA08KZziXC4ikWt/fZZScZelzsijdK/TifcQYWzoj7QvrBS7d3+UXsLiAzziUZyTAAAAAElFTkSuQmCC)
Multiplicative inverse, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAiCAMAAACOXcLLAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY625A625Bm27Zm27aQ/9uQ/9u2//+2///bagdwtQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQoU7WRyQ6AIAxEW8V9R8UF/v87rXqBSkwMcU4EOjPwAAjUjIiRtEPmhkd6dsiV0ZguaEG6LJvgTtPmVpbpJJsxE5XXgS94t9/3s/Rrmz9cpRhV9pHpBlhc6ie/xPkHgL1nbEaMH7RWh7ouOWNQgrV/JXAARpYDp59wxAYAAAAASUVORK5CYII=)
Explanation:- When a number is added to its additive inverse, the result zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for the additive inverse:-
![](data:image/png;base64,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)
(add
to both sides)
![](data:image/png;base64,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)
Solving for the multiplicative inverse:-
![](data:image/png;base64,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)
(dividing both sides by
)
![](data:image/png;base64,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)
Question 11.Write the additive and the multiplicative inverses of the following.
1
Answer:Additive inverse, -1
Multiplication inverse, 1
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
1 + x = 0
(add (-1) to both sides)
X = -1
Solving for the multiplicative inverse:-
1
X = 1
(dividing both sides by 1)
X = 1
Question 12.Write the additive and the multiplicative inverses of the following.
0
Answer:Does not exist as the answer will be 0 itself.
Question 13.Write the additive and the multiplicative inverses of the following.
![](data:image/png;base64,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)
Answer:Additive inverse, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABgAAAAhCAMAAAD03JuvAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6AGa2OgAAOgA6OmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kDqQkNv/tmYAtv//25A62////7Zm/9uQ/9u2//+2///bhMG7gwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAeElEQVQ4T8WSyQ6AIAxEC+4LIgoq2///pmK8UQ8GEudEKPPaTAFIlhfkUhVz/HqAXyTeQHf4/bsB6RAQ+QzJSXwBqBDaLRoSes5fCL++NYzQCZnADRxMPccVXWwAqo8LppFgBbbcnZCWIY7AcCP+ryxu8KLkefM6Af2VBGHnYGfaAAAAAElFTkSuQmCC)
Multiplication inverse, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAhCAMAAAAieUHKAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6AABmADo6AGa2OgAAOgA6OmZmOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kDqQkNv/tmYAtv//25A62////7Zm/9uQ/9u2//+2///bhMG7gwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAYklEQVQoU5WOSRaAIAxDg4oDiojiwHT/ayrw3KgszKYvbX9T4CnDSTECrhcw9QRdbsDKYBoFKymwE9JyFik3qFBstF5W4nUs3yBJP4i06mXAKPxywM8xH7pLo9teX+btd+IJXj0ELdOUHAMAAAAASUVORK5CYII=)
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
![](data:image/png;base64,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)
(add (
) to both sides)
![](data:image/png;base64,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)
Solving for the multiplicative inverse:-
![](data:image/png;base64,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)
(dividing both sides by
)
![](data:image/png;base64,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)
Question 14.Write the additive and the multiplicative inverses of the following.
−1
Answer:Additive inverse, -1
Multiplication inverse, 1
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
(-1) + x = 0
(add 1 to both sides)
X = 1
Solving for the multiplicative inverse:-
(-1)
X = 1
(dividing both sides by(-1))
X = 1
Question 15.Fill in the blanks.
![](data:image/png;base64,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)
Answer:let the blank be x
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Question 16.Fill in the blanks.
![](data:image/png;base64,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)
Answer:let the blank be x
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(add
on both sides)
X = 0
Question 17.Fill in the blanks.
![](data:image/png;base64,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)
Answer:let the blank be x
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 18.Fill in the blanks.
![](data:image/png;base64,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)
Answer: We know, that rational numbers are associative over addition, i.e. if
are three rational number then,
![](data:image/png;base64,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)
Therefore, on comparing we get
Question 19.Multiply
by the reciprocal of![](data:image/png;base64,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)
Answer:Reciprocal of
is ![](data:image/png;base64,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)
According to the given question
![](data:image/png;base64,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)
Question 20.Which properties can be used in computing
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAANgAAAA7CAMAAAAuLrTzAAAAAXNSR0ICQMB9xQAAAVZQTFRFgYGBAAAAiIhmZoiId3dmZnd3f39dXX9/f39dXX9/d3ddXXd3f39MTH9/ampqf25ZWW5/e2pVVWp7f25ISG5/WVl/e2pEampVVWpqRGp7f243N25/bltISFtuRllsgFszM1uAf1kzM1l/f18oKF9/bls1bls1NVtubFk1W0hIW0hISEhbbkYzM0ZuW0g1NUhbbkgdbEYdHUZsbEYdHUZsSDNIRkYzHUhbMzNbSDQ0NDRISDMzMzNIWzMdHTNbWjMdHTNaHTJaSDQeHjRIHjNHMx5HWzMAADNbWjMAADNaADJaNB00SR0dHR1JSB0dHR1IADRIHR00HR00SB0AAB1IMgAySBwAABxIMx0AAB0zMwAdMh0AAB0yMgAdHQAyHh4AAB4eHR0AKQYGKQYGMwAABgYpAAAzMwAAAAAzHQALHQALCwAdHQAAAAAdHQAAAAAdAAAAAAAAwVnA+AAAAHF0Uk5TAAEMDA0NFxcZGRoaISEkJSUmJjExMTIyMjI/P0pKS0xMTU1RUVdYWFliY2NkZHFxcnNzdHR8fX9/ioqLi4yMjY2OmZmamqCgoaGip6ioqamttre7u7u8vMnJycrKysrY2Nnb3Nzc3N3d4+Tk6+vs7P6GUz/6AAAFUUlEQVR42u2a+3fbNBTHLTlhyTroOoeOwVqnMLLx6NaEwXg1CdtoEigUCLiFFsZzLZCm8v//C8d2HOvhK1/Ja884J/qlyen1/fgrXV3pSnGcRVu0RXseGrn5jou3pm+vkYvkzahvmVMrvdtmD7T61TK6qCkvaW+qVEII0elqIjp51pJvfr+g94hWV9NgcLnPCnXlMAyP1sFne51i/7XjMG4n9eR7W/vM9Q9+0/wfw0tbOwXGbasjKFudMsbC8HXgWX8PEe8+i1s4Snt9V9Pr/q9PQvjl/a/w82t1Kgiju7wIGjxwneVxKJhw//66gSD01iRbT9sdPiwMx5vFyYH01t4eN828zUTeWb5Df4QhvKyoGTbthKF4aXc+CKTh2FHjbpg/YjQwmMn8zPL3XRthJjy/S2VhKpUGg9yHvV/q+DTNR1Ht54aNMANe7cu6Iqz2UwPp0CQyxHmli0VYmAGv13AUYc6OTIXWjqFlJOpfERaG50U0VZhMhdYdkxwl2eryIigMz/P6hJBKcLJENFHj+JtQHP9Qt4xE7aOgMDxvK1k4Q/a0DjvwPoS2OSbCpFE/Z2GzlCeHouBgIxpVsvKpWzgKBpGojSpQmAEvXxhPvTNN9nmdciCvj54uZCMckXMX1kq2eey0UQgiujJATrSwMG8S8TqFwvRFR8wYP63bZR8BFG/hzxqWoWnMWz5g7OiaaS2HpL7x1yUuO0QdPcAiXvjjlrkwjud97DrLByd1MwdYKi/s8ufV80AAPPp+NHbetPnshBHuDy9sGH63TvKMihG6ulPwJEQIJ6wYl1J/vwUfWMSPVz5piDEfTfnwaCnHqDjaNcJET1JWJMMBFlc8x+g48lJJdvsC6KW7k/BkSTUqJUz0JGbF64ffVLE4RPKIvFSCbt66Qsdp8c8blRImeuJ5dMxY+FmVM+qXzIp0/DgPlGT8NE1xRnpEtBq172vWpMhTP3fdpK1JeF81wgkjXJvnv+P9KrATaM/XscxI8EEkxBabt6ZsrHqSed78tObyn3MjkZdLdRyHcS09SAwe7c6O6BRh84KUMxJ8sKJQVEx5TwovPbHgjUQeTFUOSIOuQ2deFJD/vasYlZtjgic1QpJDjGh+6XGIrBhPHToeSaB4uEmvoRqVy4qCJ5lHg05i1C/CFQtrJSmBfiStY73H18j8FF8wKreOCZ4y3vLx0Suk0uticfab4NaEsUdX7bajW9hT64xHHzLGvl0z3ZNZbYLPZzv6rHj2m+D/ozCwgjQE8V40wuDVGs8jOZ8UKlxBGgmr3H3ClWuwME0FieVlF1HSlZRIhStIE2F3Jj+uuYhQ1FWQSF52ESVfSQlUTQVpIOw96RQDEqatING87JzLh4UJFaStsA35lEufPC5CmFBBSu3Vf1/EgWrH+1UxJVz55zXN1B8OyvEgYVf+5qlcBWl7MtsO7z0UU4Lm0ayCtOZBwmQHWQVpt5DT4Oye69ycnN1AnH0mFaRb6swOEqZQswpSKhua2Eh043HLuqet2yC3JsAhN/YaCRKmUtv5J6HIizhvOpK7R/+K3jS/I7EXf5AwlQpcaSIP02vHI+ksXXtVC995Yw/vAWE51LSCVCZPEzfH4vekwfxyW3u5nlWQ6vxrlhEmqBAqSNvY8OMfwKxmqxMciXwFWYKXK0ygChWkdZ7aOV0nK4eD4pASKsgc3u4NzBZ4I/yCSJ8iqvCrnoIK0kdG/bsTdrqZdQc4YEUVpL9XfEWQXUQJV1Iaal7b7jjGrd21L6lseFZUut00JbS6rr0wC54llW7fvkBdFjx7qulPZ52Sze6ns86iLdrz1v4DI8tLEnX2LpoAAAAASUVORK5CYII=)
Answer:![](data:image/png;base64,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)
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Multiplicative associative, multiplicative inverse, multiplicative identity, closure with addition are the properties used in computing.
Question 21.Verify the following
![](data:image/png;base64,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)
Answer:LHS:- ![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
RHS:- ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIYAAAAtCAMAAABcfOOOAAAAAXNSR0IArs4c6QAAAIRQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjoAOjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkDo6kGY6kGZmkLbbkNv/tmYAtmY6ttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///baKjBMQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACXElEQVRYR+1YaVPCMBBt8KAeQAWVilKxCj3y//+fyaZcbfaIgsM45BNDtpu3b1+S3UTReUgZ+JxLLY9kV8bKjN7PYSwnxLf661apK4HzMp7+JsBqrC4+cAdZ7y2qUkGQQTD062J/yXKwyEkYfWOfq2lUDulgg2C8j7rOSBhgnlmgvk93nIXAyG1s7cHBqGYDS6FOyeSX8V0sElEUlTc+rdEw6kSpwcpCL6jkRfrlWSYiQy6QkdmdBcMpbwdGe8oRt4wBgE49GW0pLWZNMDJ2YSAqLBR4p+mwFnXiy3rLa34N5IZqw+B3QdYJEWtm5xpDJBr3N+aF0oZLRGE2LKTTH4ebMxmuUlI+DkaBHEI5cThpe3JVabM85sF6ryZGbLClmOGNWs/sXXD5hH1b2flhk0wJ5xwKvY6JM8TndSpQIONepGLGByUOYXSHYBRO9ZDRPXrXcg/x0ral1Oz1e7IwgiPxhBfMaJeNg8HwXzz+XB8RhnfBzXWpNtN+uGs2th8E/XIHcXOqi2V+RDZOOSkIPye7YYPz6gkweLd12fiDw1yCkrna9HLMt2f01VbGgraKuegzZSrrhL666Iu+Th4FMJjSF6p2254Rg85rNqKqs41bgREjY9JDcb8SrMAU1g4sU09Q0qgf5mi1u08w0iBsjZg+lGoQoIQ3bOT884K/d9yi4Noyql0q1hcUD6NpHjENcgUvhxJvQVor0nRkfW9Tt5U4W4iKJGo3JFHgg3IKvD1kHhYsWskpClHhbyXAlIZO1D+YZxaoRqRPcJ1Hp3athMLgHp3EldDZ8N8y8A0B/C+2fE41sgAAAABJRU5ErkJggg==)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
LHS = RHS
Hence verified
Question 22.Evaluate
after rearrangement.
Answer:Given, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
(rearranging the like fractions at one place)
![](data:image/png;base64,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)
(Since +
- = -)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(L.C.M. of 5
3 = 15)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 23.Subtract
from![](data:image/png;base64,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)
Answer:Given, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAEAAAAAsCAMAAADigU7NAAAAAXNSR0IArs4c6QAAAHhQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOjqQOmaQOma2OpDbZgAAZgA6ZjoAZjo6ZmZmZpDbZrbbZrb/kDoAkGY6kLaQkLbbkNv/tmYAtpA6ttv/tv//25A627aQ2////7Zm/9uQ/9u2//+2///boKd6FAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABMklEQVRIS+WVfVPCMAzGU1ApqEzkZcgqdrQs3/8bmrXb9M4lu+r0PMgfHHe7/JInbfMAXHTYScHrK1dK3R5F/V4LADs5AJrpm0CosjUPqLIlpcZfLszS8QCvd5SH+d2JzXf3JwEQa0uA6qkAAQBmWgC+KrYDzKkCAWzdKYBRbbSqcE+H8JLNOAWuTYgAJryWhhg6kO+5eFEodQhQ8Qpi4YECpX7kD7HOpzkIGrxeHC76IV+buK8PZYwJdK9v6M8YxX5Hwhid/StG+RxWEwY3GthSPY2fVyo6kFFbOGeiG/Xp9g9H2wDqHWyVuGf7J9cAwkf3U4BsqMzRfeogmmJqfACCY6VHB8Cc9TKR2gHMTHYCjtICbG3G7hsiGovyc7qFaFIBuNe0VW42ncmnAtJn/scZ717VEoFIZvvEAAAAAElFTkSuQmCC)
(By taking L.C.M.)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC4AAAAsCAMAAAAtr3oOAAAAAXNSR0IArs4c6QAAAGlQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOmaQOpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kNv/tmYAttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bnJdhDQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA4klEQVRIS+2V3RKCIBSEpT8pUysqjEyI93/IAMlymmSZsRkvOldefCyHA7smyVTqegY7kZSYmuH4ARR2mKSTwjeULAdOyu0oXNl56OM+UQyejD/vNmY89zwFcW51JUXVuWlbsfkFVFelOXNWgfQfQybQf0zIik+me4/9j3ixMZqJ3/XXK+rSmVnXBRn0tetDFaT1AyfG13nAGzKrhMetRQUJ5pjH3V5NHM7DRn1TRzL1hWsGZEaHa4YEUofz9Abc8RMXK0M3wXZEG7lybTPY5d730if7+1rs7K26CqoD/Y6LPABX0Qsk8JE9iQAAAABJRU5ErkJggg==)
Question 24.Subtract
from 2
Answer:Given, ![](data:image/png;base64,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)
![](data:image/png;base64,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)
(Since -
- = + )
![](data:image/png;base64,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)
(By taking L.C.M.)
![](data:image/png;base64,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)
Question 25.Subtract
−7 from![](data:image/png;base64,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)
Answer:Given, ![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAFUAAAAsCAMAAAAjBgRhAAAAAXNSR0IArs4c6QAAAEVQTFRFAAAAAAAAAAA6AABmADqQAGa2OgAAOjqQOpDbZgAAZgA6ZmZmZrb/kDoAkNv/tmYAtv//25A62////7Zm/9uQ//+2///b/cd95AAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAA/UlEQVRIS+2V2xKCMAxEi5cKKtRS4P8/1YJUxQQpZvugY59gJpykyW5Q6n9iO2A3VWxofFyjE1Db4pSAanKHp7pDjae2x0ohqCYLx1+8K3PVU+05frrLkS5kgFKHvIgO0AukoSbxlu8tXrDLI/2yCHtT4K6W1N2VL/o1/XujZaom1KFEu71ISvVGZqpqC+9uyWGpo6wne2RVEo7alft5xn1jTR8eH8zV4jLZrBTbVzPKCtqBRgtnxdZqhLLyLabTEsuKpcbtYIixidIgxub1KzU2S0VMgIKT/K/eGnvV2ngOlhubSx2M/XFZ3IcAYzNYgLEpNY2s4owN7flvwK5clQivaUeesgAAAABJRU5ErkJggg==)
(Since -
- = + )
![](data:image/png;base64,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)
(By taking L.C.M.)
![](data:image/png;base64,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)
Question 26.What numbers should be added to
so as to get
.
Answer:let the unknown number be x
According to the given question,
![](data:image/png;base64,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)
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGcAAAAsCAMAAABPAFZkAAAAAXNSR0IArs4c6QAAAH5QTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOgA6OjoAOjpmOmY6OmaQOma2OpC2OpDbZgAAZjoAZjo6ZjpmZmYAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAttv/tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bSSjmPAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABw0lEQVRYR+1X2VLDMAy0A204WghXSQ+DaZuk/v8fxHKOEki8ilPe4geGmUralSxLGyGmM7gCX7vBLrWD2d9KeQ3981jaE0G7Xh4q2ooihQHy+D04Feeo5vaPlijKaJwS7OoTsL0ETrFZfKCijMc5JVIujhjnLma0y88wijrHnfr2DzGsm1mvOO0C6GbyESVEv+cxy6w/FDPAKaHmDDomJYoZ7GtFZkw6XUQMPdEinaFGUM4MXmN/rsXGTpQlghHFs+0b3P5BNZ2chlbApGiuDo3Ybe/DseNkdtQSvgQOE28+1OIXStgbxtjXplsD6O/E5CRjbfx08/h+iQM107r9T+PYSe5sS3YarnlMw1n48zm9vLYH0P/Uzax3IwZ3K1NfPoZGN3NhofL5cFK7hmnXj1x5jPtBLNulOTxxdGjjQ7I1ehsE4YyVtMIiYS8uHa2E2AcUi6lDqwRM6pSBgnu3O2GsC37hBOoQrHdrfpkts9gHzoEhYvZgZUQgTKmpeMdJKvxZ0RWsulsWDu0CGnshfaDmUE01HKqBxr/PM3tN3DJm5apMAvo6v7Hfmm4mso6WD8ewd1quLi6OcJ/LWxanyaijAt9BkhyJrU4OOQAAAABJRU5ErkJggg==)
(Add
on both the sides)
![](data:image/png;base64,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)
(By taking L.C.M.)
![](data:image/png;base64,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)
Question 27.The sum of two rational numbers is 8 If one of the numbers is
find the other.
Answer:let the unknown number be x
According to the given question,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(Add
on both the sides)
![](data:image/png;base64,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)
(By taking L.C.M.)
![](data:image/png;base64,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)
Question 28.Is subtraction associative in rational numbers? Explain with an example.
Answer:Subtraction is not associative for rational numbers because when the numbers (say a,b,c) are subtracted by grouping any two at first and the other two at second [(a-b)-c and then a-(b-c)] the answer is not same. Thus subtraction is not associative in rational numbers.
Example:- let the 3 numbers be 5,8,9
Then, at first --
(9-5)-8 = -4
And in second -
9-(5-8) = 9-(-3) = 12
Since, first case is not equal to second case’s answer
Question 29.Verify that – (–x) = x for
x = ![](data:image/png;base64,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)
Answer:According to the question,
-(-x) = x
LHS:- -(-X) RHS:- X
![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIMAAAAsCAMAAABxCftvAAAAAXNSR0IArs4c6QAAAHJQTFRFAAAAAAAAAAA6AABmADo6ADpmADqQAGa2OgAAOgA6OjpmOmaQOma2OpDbZgAAZjoAZjo6ZpDbZrbbZrb/kDoAkDo6kGY6kGZmkNv/tmYAtmY6ttv/tv//25A625Bm27aQ2////7Zm/9uQ/9u2//+2///bI0enQwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAACIUlEQVRYR+1YaVODMBANXkWrtCoesWIFmv//Fw05IIGlecvoTMdpvpSWvN2X3e0eEeK8fsECar/Jsut3rqSFMFqNzJ5EW1x+MkkshBkt6m0Xa5Mr/b3KnkVzz6HRwzggt/fjgQLVmoOgXx3TYWDsVXWnni7Z+UKVXIkGxl3NLRl8TW6010yRDsYkYbwoZObXhaGkSusg/wkKZW53UmkzqNI7iGWIAQZyttuqm28qGFb+10NBRiytQ/YwDgdahSFWW+WSJEnqCGEMErV1f7yMf5S0HMgdpIYIxuBQUXHvAtRyaHLUGREM56DKpKUXxhnO4VDQCSqUwAgIXHGwEzH0oszHYINk94oKW4aO1NZT4ICcEeGZOuux93/KYVyDZojwOYCC5w8+EeA59FUzfrCJMt2VUHDYPXw7wKKn/cDp+GLCBLCzQGzFsM5JcpjP1V99TV+Sqwd02kJRzdpvjd4m78J8SNBgzZpBpzkEtbvdZLaXGLXGWO2eQ6cp6HbS9zDNeueeRxyQ0ipm0QiHsFOjOaC9HI1GOIQ9LS0FDAdv0SVDTtDb9xzu8mHyh3t7Eo2YQUfgMOo5KepFT/6l/1/AMw6JxjgIO+t1K+yxfSTiwxuFBikEhgil+MQBmyE6AdIqR/z62d9xMLONswNj9ifQqBn0Pn/R4YqT1KHQljZvMO5ACDSDg7kLUq9dir56FKLd6oe1uR6C74JINIPCeeu/t8APR9oomCHEFz0AAAAASUVORK5CYII=)
![](data:image/png;base64,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)
(Since -
- = + )
LHS = RHS
Hence verified
Question 30.Verify that – (–x) = x for
x = ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACgAAAA1CAMAAAAEaFkaAAAAAXNSR0ICQMB9xQAAALFQTFRFgYGBAAAAiIhmZoiId3dmZnd3f39dXX9/f25Ze2pVVWp7f25ISG5/WVl/e2pEampVVWpqSFtugFszM1uAf1kzbls1bkgdHUhubEYdHUZsHUZsSDNIRkYzNDRIWzMdHTNbWjMdHTNaHTJaSDQeWzMAADNbWjMAADJaSB0dHR1IADRIHR00SB0ASBwAABxIMx0AMh0AAB0yHh4AAB4eHR0AMwAAAAAzHQAAHQAAAAAAAAAApuZkIQAAADp0Uk5TAAEMDA0NGRklJiYxMTEyMjJKTExNV3Jyc3N0fH6KjIyNjY6ZoKChoqmprba7vLzJysrY2Nnd3evs/oQiTOUAAAEDSURBVHja7ZVXc8IwEITvZFpwQjHVoSaUJISa0E7//4dlkD0CweikRybDPunhG611c7sG+O9CRH1Ahit3t7E6PK2INkUrF/3+SAWGvQAKq33efmWkQNEJTvSx5gITlTxBnE7AA8Tyepb1AMWcSL5njblpmdaicZCvZ7BJWq3rx4RHZj6XIEx9wfYycIHqgzOLln0n6vIDEQq7TQUzw4GVCw/Jq8SIiL6r8JB/6BxKQXLqMcybFmOaTLdYbieVLLHSLQZROueJK319tf/i89kB5l6S7HwFXnmGduwXfMbZBBnnqyqJ/SqZczbrjnE2QM75EmSd0xZL/hpv4GwxpXHtfpb6DwGsJlNXS0pzAAAAAElFTkSuQmCC)
Answer:According to the question,
-(-x) = x
LHS:- -(-X) RHS:- X
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(Since -
- = + )
LHS = RHS
Hence verified
Question 31.Write-
(i) The set of numbers which do not have any additive identity
(ii) The rational number that does not have any reciprocal
(iii) The reciprocal of a negative rational number.
Answer:(i) Natural numbers
(ii) 0(zero) is the rational number which does not have a reciprocal.
(iii) Is a negative rational number
The reciprocal of a negative number must itself be a negative number so that the number and its reciprocal multiply to 1.
Example:- ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAhCAMAAAAIybBlAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAdklEQVQoU62RSw6AIAxEW8Uf/lFQFO9/TRETA11B4ixfp2VaAJIkS8w6v+MaZ1CZIEOOgpBz6q3FNOg0ACyYPyTQxmzhk+ECjoCAZOT1pOwx5jefp5imnz17i1j5Mw2PujGsNUmigjG2uFKgLdDB1d0X+SR1uxu9cQOujzMsIAAAAABJRU5ErkJggg==)
Reciprocal ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABEAAAAiCAMAAACOXcLLAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAd0lEQVQoU62QRw6AMAwE7RBa6C208P9vEkBIsTmAFPZkjdbrAuCpKUOM3QyjGtCiZbFLyEmfWItJ8VRpa01iLOg5mC2YD+utq90lntc826/9HP0+4T1wiFDkrm2rPv14rQsa3mHACMAoyUdVCwshMEg2/X1h6tgBYFcDri034Y0AAAAASUVORK5CYII=)
Name the property involved in the following examples
Answer: Role of zero/additive identity:
0 is the additive identity for rational numbers, i.e. if 'a' is any rational number then
a + 0 = a = 0 + a
Question 2.
Name the property involved in the following examples
Answer:
Distributive law
This property is know as the distributive law of multiplication over addition.
For all rational numbers a, b and c
a(b + c) = ab + ac
⇒
Question 3.
Name the property involved in the following examples
Answer:
Multiplicative identity
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.
Question 4.
Name the property involved in the following examples
Answer:
Multiplicative identity
The multiplicative identity property states that any time you multiply a number by 1, the result, or product, is that original number.
Question 5.
Name the property involved in the following examples
Answer:
Commutative law of addition
In commutative law of addition, a, b is rational number where
a + b = b + a
Question 6.
Name the property involved in the following examples
Answer:
Closure law in multiplication
We find that rational numbers are closed under multiplication. For any two rational numbers a and b, ab is also rational number.
Question 7.
Name the property involved in the following examples
7a + (-7a) = 0
Answer:
(vii) Additive inverse law
Any two numbers whose sum is 0 are called the additive inverses of each other. In general if ‘a’ represents any rational number then a + (-a) = 0 and (-a ) + a = 0
Then a, (-a) are additive inverse of each other.
Question 8.
Name the property involved in the following examples
Answer:
(viii) Multiplicative inverse
We say that a rational number is called the reciprocal or the multiplicative inverse of another rational number
if
Question 9.
Name the property involved in the following examples
Answer:
Distributive property
This property is known as distribution law of multiplication over addition. For all rational numbers a, b and c
a + b = b + a
Question 10.
Write the additive and the multiplicative inverses of the following.
Answer:
Additive inverse,
Multiplicative inverse,
Explanation:- When a number is added to its additive inverse, the result zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for the additive inverse:-
(add to both sides)
Solving for the multiplicative inverse:-
(dividing both sides by )
Question 11.
Write the additive and the multiplicative inverses of the following.
1
Answer:
Additive inverse, -1
Multiplication inverse, 1
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
1 + x = 0
(add (-1) to both sides)
X = -1
Solving for the multiplicative inverse:-
1X = 1
(dividing both sides by 1)
X = 1
Question 12.
Write the additive and the multiplicative inverses of the following.
0
Answer:
Does not exist as the answer will be 0 itself.
Question 13.
Write the additive and the multiplicative inverses of the following.
Answer:
Additive inverse,
Multiplication inverse,
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
(add () to both sides)
Solving for the multiplicative inverse:-
(dividing both sides by )
Question 14.
Write the additive and the multiplicative inverses of the following.
−1
Answer:
Additive inverse, -1
Multiplication inverse, 1
Explanation:- When a number is added to its additive inverse, the result is zero. When a number is multiplied to its multiplicative inverse, the result is 1.
Solving for additive inverse:-
(-1) + x = 0
(add 1 to both sides)
X = 1
Solving for the multiplicative inverse:-
(-1)X = 1
(dividing both sides by(-1))
X = 1
Question 15.
Fill in the blanks.
Answer:
let the blank be x
⇒
⇒
⇒
Question 16.
Fill in the blanks.
Answer:
let the blank be x
(add on both sides)
X = 0
Question 17.
Fill in the blanks.
Answer:
let the blank be x
Question 18.
Fill in the blanks.
Answer: We know, that rational numbers are associative over addition, i.e. if
Therefore, on comparing we get
Question 19.
Multiply by the reciprocal of
Answer:
Reciprocal of is
According to the given question
Question 20.
Which properties can be used in computing
Answer:
Multiplicative associative, multiplicative inverse, multiplicative identity, closure with addition are the properties used in computing.
Question 21.
Verify the following
Answer:
LHS:-
RHS:-
LHS = RHS
Hence verified
Question 22.
Evaluateafter rearrangement.
Answer:
Given,
(rearranging the like fractions at one place)
(Since + - = -)
(L.C.M. of 53 = 15)
Question 23.
Subtractfrom
Answer:
Given,
(By taking L.C.M.)
Question 24.
Subtractfrom 2
Answer:
Given,
(Since - - = + )
(By taking L.C.M.)
Question 25.
Subtract
−7 from
Answer:
Given,
(Since - - = + )
(By taking L.C.M.)
Question 26.
What numbers should be added toso as to get
.
Answer:
let the unknown number be x
According to the given question,
(Add on both the sides)
(By taking L.C.M.)
Question 27.
The sum of two rational numbers is 8 If one of the numbers is find the other.
Answer:
let the unknown number be x
According to the given question,
(Add on both the sides)
(By taking L.C.M.)
Question 28.
Is subtraction associative in rational numbers? Explain with an example.
Answer:
Subtraction is not associative for rational numbers because when the numbers (say a,b,c) are subtracted by grouping any two at first and the other two at second [(a-b)-c and then a-(b-c)] the answer is not same. Thus subtraction is not associative in rational numbers.
Example:- let the 3 numbers be 5,8,9
Then, at first --(9-5)-8 = -4
And in second - 9-(5-8) = 9-(-3) = 12
Since, first case is not equal to second case’s answer
Question 29.
Verify that – (–x) = x for
x =
Answer:
According to the question,
-(-x) = x
LHS:- -(-X) RHS:- X
(Since - - = + )
LHS = RHS
Hence verified
Question 30.
Verify that – (–x) = x for
x =
Answer:
According to the question,
-(-x) = x
LHS:- -(-X) RHS:- X
(Since - - = + )
LHS = RHS
Hence verified
Question 31.
Write-
(i) The set of numbers which do not have any additive identity
(ii) The rational number that does not have any reciprocal
(iii) The reciprocal of a negative rational number.
Answer:
(i) Natural numbers
(ii) 0(zero) is the rational number which does not have a reciprocal.
(iii) Is a negative rational number
The reciprocal of a negative number must itself be a negative number so that the number and its reciprocal multiply to 1.
Example:-
Reciprocal
Exercise 1.2
Question 1.Represent these numbers on the number line.
(i) 9/7 (ii) -7/5
Answer:In a rational number, the number below the bar i.e. the denominator tells the number of equal parts into which the first unit has been divided. The numerator tells ‘how many’ of these parts are considered.
(i) Here
means 9 markings of
each on the right of zero and starting from 0. The 9th marking is
.
![](data:image/jpeg;base64,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)
(ii) Here
means 7 markings of
each on the left of zero and starting from 0. The 7th marking is
.
![](data:image/jpeg;base64,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)
Question 2.Write five rational numbers which are smaller than
.
Answer:Now we have to write 5 numbers which are less than ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAiCAMAAACk7TNkAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6ADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgA6ZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmY625A627Zm29u229vb2////9uQ/9u2//+2///beWS0qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQoU41PSRKAIAwrioK7ggso/3+nARw9KDPm0E7Tpk2JUpgZY5kimoc4cWfwMlK2iD031uQmFWqnIeuTW18NbPP4L3gml04YHNcVIvzxy4voSvC2kNs6cjoa/0MOj9I41LS38B8UXzgB4qMEMFCLZAQAAAAASUVORK5CYII=)
It is very simple.
![](data:image/png;base64,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)
Therefore
are 5 numbers which are less than ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAiCAMAAACk7TNkAAAAAXNSR0IArs4c6QAAAFdQTFRFAAAAAAAAAAA6ADqQAGa2OgAAOjo6OmaQOma2OpC2OpDbZgA6ZjoAZjo6ZrbbZrb/kDoAkDo6kNv/tmY625A627Zm29u229vb2////9uQ/9u2//+2///beWS0qAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAa0lEQVQoU41PSRKAIAwrioK7ggso/3+nARw9KDPm0E7Tpk2JUpgZY5kimoc4cWfwMlK2iD031uQmFWqnIeuTW18NbPP4L3gml04YHNcVIvzxy4voSvC2kNs6cjoa/0MOj9I41LS38B8UXzgB4qMEMFCLZAQAAAAASUVORK5CYII=)
Question 3.Find 12 rational numbers between -1 and 2.
Answer:let us multiply and divide (-1) and 2 by 12
![](data:image/png;base64,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)
![](data:image/png;base64,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)
12 rational numbers are,
![](data:image/png;base64,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)
Question 4.Find a rational number between
and
.
[Hint: First write the rational numbers with equal denominators.]
Answer:make denominators same
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Therefore
lies between
and ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAcAAAAhCAMAAAAieUHKAAAAAXNSR0IArs4c6QAAAGNQTFRFAAAAAAAAAAA6AABmADo6ADqQOgAAOgA6OjoAOma2ZgA6ZjpmZpDbZrbbZrb/kDoAkDo6kGY6kGaQkLbbkNv/tmYAtv//25A625Bm27Zm27aQ29uQ/7Zm/9uQ/9u2//+2///bBXbt7gAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAZElEQVQoU42OSQ6AIBRDiwOOiCM4y/1PqdSNMTGxm7do/2+Bt8ZUBCXg6g5T0NPdEnJvlMcgIhKYZYUj77FdxCh591fi1t/4M8c6vzAlrdKea+Y8j8I4rVpYvo8ZpA8sMjRffSc4jARN8P2UugAAAABJRU5ErkJggg==)
Question 5.Find ten rational numbers between
and
.
Answer:make denominators same
![](data:image/png;base64,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)
![](data:image/png;base64,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)
10 rational numbers are,
![](data:image/png;base64,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)
Represent these numbers on the number line.
(i) 9/7 (ii) -7/5
Answer:
In a rational number, the number below the bar i.e. the denominator tells the number of equal parts into which the first unit has been divided. The numerator tells ‘how many’ of these parts are considered.
(i) Here means 9 markings of
each on the right of zero and starting from 0. The 9th marking is
.
(ii) Here means 7 markings of
each on the left of zero and starting from 0. The 7th marking is
.
Question 2.
Write five rational numbers which are smaller than.
Answer:
Now we have to write 5 numbers which are less than
It is very simple.
Therefore
are 5 numbers which are less than
Question 3.
Find 12 rational numbers between -1 and 2.
Answer:
let us multiply and divide (-1) and 2 by 12
12 rational numbers are,
Question 4.
Find a rational number betweenand
.
[Hint: First write the rational numbers with equal denominators.]
Answer:
make denominators same
Therefore lies between
and
Question 5.
Find ten rational numbers betweenand
.
Answer:
make denominators same
10 rational numbers are,
Exercise 1.3
Question 1.Express each of the following decimal in the
form.
(i) 0.57
(ii) 0.176
(iii) 1.00001
(iv) 25.125
Answer:(i) ![](data:image/png;base64,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)
(ii) ![](data:image/png;base64,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)
(iii) ![](data:image/png;base64,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)
(iv) ![](data:image/png;base64,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)
Question 2.Express each of the following decimals in the rational form
.
![](data:image/png;base64,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)
Answer:let x = ![](data:image/png;base64,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)
x = 0.99999….. -→(i)
Here the periodicity of the decimal is one
So, we multiply both sides of (i) by 10 and we get
10x = 9.999…-→(ii)
Subtract (i) from (ii)
10x = 9.999….
x = 0.999…
10x - x = 9.999... - 0.999...
9x = 9.0
x = 1
Hence
= 1
Question 3.Express each of the following decimals in the rational form
.
![](data:image/png;base64,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)
Answer:let x = ![](data:image/png;base64,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)
X = 0.575757….. -→(i)
Here the periodicity of the decimal is two
So, we multiply both sides of (i) by 100 and we get
100x = 57.575757…-→(ii)
Subtract (i) from (ii)
100x = 57.5757….
X = 0.5757…
99x = 57.0
![](data:image/png;base64,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)
(divide by 99 on both sides)
![](data:image/png;base64,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)
Hence
=![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAiCAMAAAC3OnOQAAAAAXNSR0IArs4c6QAAAGxQTFRFAAAAAAAAAAA6AABmADo6AGaQAGa2OgAAOgA6OjoAOmZmOpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDoAkDo6kGY6kLbbkNv/tmYAtmY6tv//25A625Bm27Zm27aQ2////7Zm/9uQ/9u2//+2///bBI4rJgAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAmElEQVQoU6WR3RaCIBCEZ+3HsiKzgpJKkPd/x9gFzTuPJy74WHYHlgFYPMJjyxqnqDhH2kpx3B9ruPLCmZbjbvUEzGGM3U7DN6KUPFqiSv3yvNWfdJyD2XykKS/pWEcUF6FZ14t7nRfw6ZMxL/i3wu6zb4nheser0CPF2ujVhP6WHp5pKBkxEHgn34XsIf/DQNgy35e5rP8v+7UItfNMI3UAAAAASUVORK5CYII=)
Question 4.Find (x + y) ÷ (x − y) if
![](data:image/png;base64,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)
Answer:According to the question we have,
(x + y) ÷ (x − y)
Put the values of x and y
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 5.Find (x + y) ÷ (x − y) if
x =
, y = ![](data:image/png;base64,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)
Answer:According to the question we have,
(x + y) ÷ (x − y)
Put the values of x and y
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 6.Divide the sum of
and
by the product of
and
.
Answer:To find the sum,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
To find the sum,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
According to the given question,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 7.If
of a number exceeds
of the same number by 36. Find the number.
Answer:let the number is x
Then
of x ![](data:image/png;base64,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)
According to the question,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
9x = 36
35
(multiply 35 on both the sides)
9X = 1260
![](data:image/png;base64,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)
(divide by 9 on both the sides)
X = 140
Question 8.Two pieces of lengths
m and
m are cut off from a rope 11 m long. What is the length of the remaining rope?
Answer:total length of the rope = 11m
Let the third part of rope be x
Length of first piece =
m =
m
Length of second piece =
m =
m
According to the question,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(By taking L.C.M.)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
Question 9.The cost of
meters of cloth is
. Find the cost per metre.
Answer:cost of cloth =
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAiCAMAAAC3OnOQAAAAAXNSR0IArs4c6QAAAF1QTFRFAAAAAAAAAAA6AABmADqQAGaQAGa2OgAAOjoAOmZmOma2OpDbZjoAZjpmZrbbZrb/kDoAkDo6kGaQkNv/tmYAtmY6tv//25A625Bm29uQ2////7Zm/9uQ//+2///bL3fpFQAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAhUlEQVQoU62QwRKDIAxEpbW1xRZBqqi1+f/PdIXIwKUzOu7tkc2GpCj2ygkhLhZd1N3XXteEhOH5yhiVwPBL7wgMzZXviUy6jkzGcp0+t2mdgnGKc4T3nSr8LtWp2X/C+ASbg8yDzxhenNIpfyWl/Hv3pFWL67Dbb1BGxmvmB4/VtT+y2wJOYgWgl+XLTAAAAABJRU5ErkJggg==)
Length of cloth =
= ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAA8AAAAiCAMAAAC3OnOQAAAAAXNSR0IArs4c6QAAAFpQTFRFAAAAAAAAAAA6ADo6OgAAOgA6OjoAOma2OpDbZgAAZgA6ZjoAZpDbZrbbZrb/kDo6kGY6kLbbkNv/tmYA25A625Bm27Zm27aQ29u2/7Zm/9uQ/9u2//+2///bQHBKxAAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAAAgklEQVQoU62QyQ6DMBBDPSwFurCkrIH+/292MskAlXppFV8sOxpHesCvmm9EBR8NF0ruwHbtMCYGr8a7k80/vS+lXtuH+OjOgSelknsfWVNWAwvHpeZdA8t5q4jF/ZDJf7Hl1k+KPf99L7DTxzM77ZRhyMouRGV3zAu7IGW3F/+zewOc3wVG9+YYQAAAAABJRU5ErkJggg==)
Cost per meter = ![](data:image/png;base64,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)
(divide cost of cloth by length of cloth)
= 1.66 is the cost per meter
Question 10.Find the area of a rectangular park which is
m long and
m broad.
Answer:length =
m = ![](data:image/png;base64,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)
Breadth =
m = ![](data:image/png;base64,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)
Area = length
breadth
= ![](data:image/png;base64,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)
Question 11.What number should
be divided by to get
?
Answer:let the number which be divided by be x
According to the question
![](data:image/png;base64,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)
![](data:image/png;base64,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)
(Multiply by x on both the sides)
![](data:image/png;base64,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)
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Question 12.If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is the length of the cloth required for each trouser?
Answer:number of trousers = 36
Length of cloth = 64m
Length required for each trouser =
m
Question 13.When the repeating decimal 0.363636 .... is written in simplest fractional form
, find the sum p + q.
Answer:x = 0.363636…(i)
Periodicity = 2
So,
100x = 36.363636….(ii)
From (i) and (ii)
100x = 36.363636….
X = 0.3636363…
_____________________
99x = 36
X = ![](data:image/png;base64,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)
Then p = 4 and q = 11
p + q = 4 + 11 = 15
Express each of the following decimal in theform.
(i) 0.57
(ii) 0.176
(iii) 1.00001
(iv) 25.125
Answer:
(i)
(ii)
(iii)
(iv)
Question 2.
Express each of the following decimals in the rational form .
Answer:
let x =
x = 0.99999….. -→(i)
Here the periodicity of the decimal is one
So, we multiply both sides of (i) by 10 and we get
10x = 9.999…-→(ii)
Subtract (i) from (ii)
10x = 9.999….
x = 0.999…
10x - x = 9.999... - 0.999...9x = 9.0
x = 1
Hence = 1
Question 3.
Express each of the following decimals in the rational form .
Answer:
let x =
X = 0.575757….. -→(i)
Here the periodicity of the decimal is two
So, we multiply both sides of (i) by 100 and we get
100x = 57.575757…-→(ii)
Subtract (i) from (ii)
100x = 57.5757….
X = 0.5757…
99x = 57.0
(divide by 99 on both sides)
Hence =
Question 4.
Find (x + y) ÷ (x − y) if
Answer:
According to the question we have,
(x + y) ÷ (x − y)
Put the values of x and y
Question 5.
Find (x + y) ÷ (x − y) if
x = , y =
Answer:
According to the question we have,
(x + y) ÷ (x − y)
Put the values of x and y
Question 6.
Divide the sum of and
by the product of
and
.
Answer:
To find the sum,
To find the sum,
According to the given question,
Question 7.
If of a number exceeds
of the same number by 36. Find the number.
Answer:
let the number is x
Then of x
According to the question,
9x = 3635
(multiply 35 on both the sides)
9X = 1260
(divide by 9 on both the sides)
X = 140
Question 8.
Two pieces of lengths m and
m are cut off from a rope 11 m long. What is the length of the remaining rope?
Answer:
total length of the rope = 11m
Let the third part of rope be x
Length of first piece = m =
m
Length of second piece = m =
m
According to the question,
(By taking L.C.M.)
Question 9.
The cost of meters of cloth is
. Find the cost per metre.
Answer:
cost of cloth = =
Length of cloth = =
Cost per meter =
(divide cost of cloth by length of cloth)
= 1.66 is the cost per meter
Question 10.
Find the area of a rectangular park which ism long and
m broad.
Answer:
length = m =
Breadth = m =
Area = length breadth
=
Question 11.
What number shouldbe divided by to get
?
Answer:
let the number which be divided by be x
According to the question
(Multiply by x on both the sides)
Question 12.
If 36 trousers of equal sizes can be stitched with 64 meters of cloth. What is the length of the cloth required for each trouser?
Answer:
number of trousers = 36
Length of cloth = 64m
Length required for each trouser = m
Question 13.
When the repeating decimal 0.363636 .... is written in simplest fractional form, find the sum p + q.
Answer:
x = 0.363636…(i)
Periodicity = 2
So,
100x = 36.363636….(ii)
From (i) and (ii)
100x = 36.363636….
X = 0.3636363…
_____________________
99x = 36
X =
Then p = 4 and q = 11
p + q = 4 + 11 = 15