OPERATIONS WITH FRACTIONS

Problem 1 :

The sum of 3/5, 2/3 and 1/4 is

A) 1/2     B) 27/20     C) 3/2     D) 91/60     E) 1 5/12

Solution :

3/5 + 2/3 + 1/4

Find the least common multiple of the denominators.

LCM of 5, 3 and 4 is 60

Make the denominator of each fraction as 60 using multiplication.

3/5 = (3 × 12) / (5 × 12) = 36/60

2/3 = (2 × 20) / (3 × 20) = 40/60

1/4 = (1 × 15) / (4 × 15) = 15/60

Then, we have

3/5 + 2/3 + 1/4 = 36/60 + 40/60 + 15/60

3/5 + 2/3 + 1/4 = (36 + 40 + 15 / 60)

3/5 + 2/3 + 1/4 = 91/60

So, option (D) is correct.

Problem 2 :

Subtract 3/4 from 9/10.

A)  3/20     B)  1     C) 3/5     D) 3/40     E) 7/40

Solution :

9/10 - 3/4

Find the least common multiple of the denominators.

LCM of 10, 4 is 20

Make the denominator of each fraction as 20 using multiplication.

9/10 = (9 × 2) / (10 × 2) = 18/20

3/4 = (3 × 5) / (4 × 5) = 15/20

Then, we have

9/10 - 3/4 = 18/20 - 15/20

9/10 - 3/4 = (18 - 15) / 20

9/10 - 3/4 = 3/20

So, option (A) is correct.

Problem 3 :

5/6 ÷ (4/3 ∙ 5/4) is equal to

A) 2     B) 50/36     C) 1/2     D) 36/50     E) 7/12

Solution :     

= 5/6 ÷ (4/3 ∙ 5/4)

= 5/6 ÷ (5/3)

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

= 5/6 × 3/5

5/6 ÷ (4/3 ∙ 5/4) = 1/2

So, option (C) is correct.

Problem 4 :

Subtract 32 3/5 from 57.

A) 24 2/5     B) 25 3/5     C) 25 2/5     D) 24 3/5     E) 24 1/5

Solution :

= 57 - 32 3/5

Convert mixed fraction into fraction.

= 57 - 163/5

= (285 - 163) / 5

= 122/5

57 - 32 3/5 = 24 2/5

So, option (A) is correct.

Problem 5 :

Divide 4 1/2 by 1 1/8.

A) 1/4     B) 4     C) 8/9     D) 9/8     E) 3 1/2

Solution : 

= 4 1/2 ÷ 1 1/8

Convert mixed fraction into fraction.

= (9/2) ÷ (9/8)

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

(9/2) ÷ (9/8) = (9/2) × (8/9)

(9/2) ÷ (9/8) = 4

So, option (B) is correct.

Problem 6 :

Which of the following fractions is the largest?

A)1/2     B)11/16     C)5/8     D)21/32     E)3/4

Solution :

Find the least common multiple of the denominators.

LCM of 2, 16, 8, 32 and 4 is 32

Make the denominator of each fraction as 32 using multiplication.

1/2 = (1 × 16) / (2 × 16) = 16/32

11/16 = (11 × 2) / (16 × 2) = 22/32

5/8 = (5 × 4) / (8 × 4) = 20/32

21/32 = (21 × 1) / (32 × 1) = 21/32

3/4 = (3 × 8) / (4 × 8) = 24/32

Largest fraction:

24/32 > 22/32 > 21/32 > 20/32 > 16/32

3/4 > 11/16 > 21/32 > 5/8 > 1/2

So, option (E) is correct.

Problem 7 :

Which of the following fractions is closest to 2/3?

A) 11/15    B) 7/10     C) 4/5     D) 1/2     E) 5/6

Solution : 

11/15 - 2/3 = (11/15) - (10/15) = 1/15

7/10 - 2/3 = (21/30) - (20/30) = 1/30

4/5 - 2/3 = (12/15) - (10/15) = 2/15

1/2 - 2/3 = (3/6) - (4/6) = -1/6

5/6 - 2/3 = (5/6) - (4/6) = 1/6

Since 1/30 is closest to zero, 7/10 is closest to 2/3.

So, option (B) is correct.

Problem 8 :

Simplify (4 - 9/10) / (2/3 + 1/2)

A) 93/5     B) 93/35     C) 147/35     D) 147/5     E) 97/35

Solution :

= (4 - 9/10) / (2/3 + 1/2)

4 - 9/10 = (40 - 9) / 10 = 31/10

2/3 + 1/2

LCM of 3, 2 is 6.

2/3 = (2 × 2) / (3 × 2) = 4/6

1/2 = (1 × 3) / (2 × 3) = 3/6

2/3 + 1/2 = 4/6 + 3/6

Then, we have

2/3 + 1/2 = 7/6

(4 - 9/10) / (2/3 + 1/2) = (31/10) / (7/6)

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

(4 - 9/10) / (2/3 + 1/2) = (31/10) × (6/7)

(4 - 9/10) / (2/3 + 1/2) = 93/35

So, option (B) is correct.

Problem 9 :

Find the value of (1/a + 1/b) / (1/a - 1/b) when a = 3, b = 4.

A)7     B)2     C)1     D)1/7     E)2/7

Solution :

= (1/a + 1/b) / (1/a - 1/b)

= (1/3 + 1/4) / (1/3 - 1/4)

= (7/12) / (1/12)

To divide a fraction by another fraction, multiply the first fraction by the reciprocal of the second fraction.

(1/3 + 1/4) / (1/3 - 1/4) = (7/12) × 12

(1/3 + 1/4) / (1/3 - 1/4) = 7

So, option (A) is correct.