Exercise 1.3

1. Express each of the following decimal in the pq form.

i. 0.57

Solution:

ii. 0.176

Solution:

iii. 1.00001

Solution:

iv. 25.125

Solution: =

2. Express each of the following decimals in the rational form (pq)

i. 0.9

Solution: .

ii. 0.57

Solution:

iii. 0.729

Solution:

iv. 12.28

Solution: 1228 ÷ 4 = while 100 ÷ 4 =

So, 12.28 in simplest rational form is .

3. Find (x + y) ÷ (x - y) if:

i. x = 52 , y = −34

Solution:

Here's how to find (x + y) ÷ (x - y) when x = 5/2 and y = -3/4:

Calculate x + y:

x + y = (5/2) + (-3/4)

To add these fractions, find a common denominator (which is 4):

x + y = + =

Calculate x - y:

x - y = (5/2) - (-3/4)

Again, use a common denominator of 4:

x - y = (10/4) - (-3/4) =

Divide (x + y) by (x - y):

(x + y) ÷ (x - y) = (7/4) ÷ (13/4)

To divide fractions, multiply by the reciprocal of the second fraction:

(7/4) × (4/13) =

Therefore, (x + y) ÷ (x - y) = .

ii. x= 14 , y= −32

Solution:

Here's how to calculate (x + y) ÷ (x - y) when x = 1/4 and y = -3/2:

Calculate x + y:

x + y = (1/4) + (-3/2)

To add these fractions, find a common denominator (which is 4):

x + y = (1/4) + (-6/4) =

Calculate x - y:

x - y = (1/4) - (-3/2)

Again, use a common denominator of 4:

x - y = (1/4) - (-6/4) =

Divide (x + y) by (x - y):

(x + y) ÷ (x - y) = (-5/4) ÷ (7/4)

To divide fractions, multiply by the reciprocal of the second fraction:

(-5/4) × (4/7) =

Therefore, (x + y) ÷ (x - y) = -5/7

4. Divide the sum of -13/5 and 12/7 by the product of -13/7 and-1/2.

Solution:

Find the sum of -13/5 and 12/7:

To add these fractions, find a common denominator :

(-13/5) + (12/7) = + =

Find the product of -13/7 and -1/2:

Multiply the numerators and the denominators:

(-13/7) × (-1/2) = (Remember, a negative times a negative is a )

Divide the sum by the product:

Divide the sum (-31/35) by the product (13/14). To divide fractions, multiply by the reciprocal of the second fraction:

−3135 ÷ 1314 = −3135 × 1413 = −31×1435×13 =

Simplify (if possible):

Both 434 and 455 are divisible by :

Find the sum of −135 and 127:

To add these fractions, find a common denominator :

−135 + 127 = −9135 + 6035 =

Find the product of −137 and −12:

Multiply the numerators and the denominators:

−137 × −12 = (Remember, a negative times a negative is a )

Divide the sum by the product:

Divide the sum −3135 by the product 1314. To divide fractions, multiply by the reciprocal of the second fraction:

−3135 ÷ 1314 = −3135 × 1413 = −31×1435×13 =

Simplify (if possible):

Both 434 and 455 are divisible by :

−4347 =

4557 =

So, the simplified answer is .

5. If 25 of a number exceeds 17 of the same number by 36. Find the number.

Solution: Let "x" represent the unknown number. The problem can be translated into the following equation: x - x =

The least common denominator for 5 and 7 is .

Rewrite the fractions with this denominator: x - x = i.e. x =

Multiply both sides of the equation by the reciprocal of 935, which is : x = 36 × 359

x = 369 × 35 = × 35 =

Therefore, the number is 140.

6. Two pieces of lengths 235m and 3310m are cut off from a rope 11 m long. What is the length of the remaining rope?

Solution:

2 35 = 2×5+35 =

3 310 = 3×10+310 =

Add the two fractions: 135 + 3310

Find a common denominator i.e. : 2610 + 3310 =

Subtracting: -

Converting 11 to a fraction with a denominator of 10:

So: - =

Upon further simplification: 5110 = 51101 510

The length of the remaining rope is 5 110 meters.

7. The cost of 723 meters of cloth is 4 ₹ 1234.Find the cost per metre.

Solution:

Cloth length: 7 23 meters = 7×3+23 = 233255 meters

Cloth cost: 4 124 rupees = 4×4+124 = 284244 rupees = rupees

Cost per meter = TotalcostTotallength

Cost per meter = 7rupees233meters

To divide, multiply by the reciprocal: 7 × 323233 = rupees per meter

The cost per meter of cloth is 2123 rupees, which is approximately ₹ .

8. Find the area of a rectangular park which is 1835m long and 823m board.

Solution:

Area = ×

Area = 935 × 263

Area = 93×265×3

Area = /

Both 2418 and 15 are divisible by :

24183 =

153 =

Area =

806 divided by is with a remainder of .

So, the mixed number is 161 15156 12.

The area of the rectangular park is 161 15 m2.

9. What number should −1316 be divided by to get −114?

Solution:.

Let the number we need to find be represented by 'x'. The problem can be written as an equation:

−1316x =

To solve for x, we can multiply both sides of the equation by x:

−1316 = −114 × x

Now, to isolate x, we can multiply both sides by the reciprocal of −114, which is :

x = −1316 × −411

Multiply the fractions:

x = 13×416×11

x =

Simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 4:

x = 52÷4176÷4

x =

Therefore, −1316 should be divided by 1344 to get −114.

10. 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?

Solution:.

If 36 trousers require 64 meters of cloth, then the cloth required for one trouser can be found by dividing the total cloth by the number of trousers:

64 meters36 trousers = meters per trouser

We can also express this as a fraction:

6436 = meters per trouser

So, each trouser requires 169 meters of cloth, or approximately meters of cloth.

11. When the repeating decimal 10.363636... is written in simplest fractional form pq find the value of p + q.

Solution:

Let x equal the repeating decimal: x = 10.363636...

x = + 0.363636...

Let y = 0.363636...

Thus, y = 36.363636...

So: 100y - y = 36.363636... - 0.363636...

y = 36

y =

Both numerator and denominator are divisible by 9:

y = 36÷999÷9 =

x = 10 + 411 = + (4/11) =

Therefore, the simplest fractional form is 11411.

Now, to find p + q:

p =

q =

p + q = 114 + 11 =

So, the value of p + q is 125.