FRACTION WORD PROBLEMS - NUMBER OF SOLVED EXAMPLES AND EXERCISES.

Please study
Fractions before Fraction Word Problems.

The knowledge of converting mixed fractions to improper fractions
and improper fractions to mixed fractions is a prerequisite here.

Learn/Teach
Fractions through Fun Games.
For details, see near
the bottom of this page.

Also required is the knowledge of

Adding Fractions,

Subtracting Fractions,

Multiplyingfractions

and

Dividing Fractions.

We also need to know the method of finding
Least Common Multiple (L.C.M.).
The idea of solving
linear equations in one variable
is used in some problems.

Understanding the word statements of the problem and
writing the word statements in the solution along with
the method of solution are explained in a lucid way.

Solved Example 1 of Fraction Word Problems

Three boxes weigh 10  ¹⁄₂ kg, 15  ¹⁄₅ kg and 20  ³⁄₄ kg respectively. A porter carries all the three boxes. What is the total weight carried by the porter ?

Solution :
Total weight
= Sum of the weights of all the three boxes
= 10  ¹⁄₂ + 15  ¹⁄₅ + 20  ³⁄₄

= (10 x 2 + 1)⁄2 + (15 x 5 + 1)⁄5 + (20 x 4 + 3)⁄4 = 21⁄2 + 76⁄5 + 83⁄4

Let us find the L.C.M. of the denominators.

        2|    2     4     5
        -----------------------
              1     2     5
        

Solved Example 2 of Fraction Word Problems

From a rope of 15  ³⁄₄ m long, a piece of length 6  ⁵⁄₆ m has been cut off. What is the length of the remaining piece ?

Solution :
Length of the remaining piece
= Total length of the rope - Length of cut piece
= 15  ³⁄₄ m - 6  ⁵⁄₆ m
= (15 x 4 + 3)⁄4 - (6 x 6 + 5)⁄6
= 63⁄4 - 41⁄6

        2|    4     6     
        ----------------
              2     3    
        

Solved Example 3 of Fraction Word Problems

The weight of 15 packets of sweets is 7  ¹⁄₂ kg. What is the weight of each packet.

Solution :
Weight of each packet
= weight of 15 packets of sweets ÷ 15
= (7  ¹⁄₂) ÷ 15
= [(7 x 2 + 1)⁄2] ÷ 15 = (15⁄2) ÷ 15
= (15⁄2) x (1⁄15) [ since dividing is same as multiplying with reciprocal ]
= 1⁄2 kg. Ans.

Solved Example 4 of Fraction Word Problems

An iron rod is 9  ¹⁄₂ metres long and it is cut into 19⁄20 metre long pieces. How many pieces are made out of the long rod ?

Solution :
Number of pieces of the rod
= (Total length of the rod) ÷ (length of each piece of rod)
= (9  ¹⁄₂) ÷ (19⁄20)
= [(9 x 2 + 1)⁄2] ÷ (19⁄20)
= (19⁄2) ÷ (19⁄20)
= (19⁄2) x (20⁄19) [ since dividing is same as multiplying with reciprocal ]
= 10. Ans.

Solved Example 5 of Fraction Word Problems

A mango weighs 1⁄3 kilo. If a basket of mangoes weighs 10  ²⁄₃ kg., find the number of mangoes in the basket (weight of basket is ignored).

Solution :
Number of mangoes in the basket
= (Total weight of mangoes in the basket) ÷ (weight of each mango)
= (10  ²⁄₃) ÷ (1⁄3)
= [(10 x 3 + 2)⁄3] ÷ (1⁄3)
= (32⁄3) ÷ (1⁄3)
= (32⁄3) x (3⁄1) [ since dividing is same as multiplying with reciprocal]
= 32. Ans.

The Examples 6, 7, 8 and 10
need the knowledge of solving

Linear Equations in one variable

Solved Example 6 of Fraction Word Problems

A student was asked to multiply a given number by 9⁄16. Instead he divided the given number by 9⁄16. His answer was 175 more than the correct answer. What was the given number.

Solution to fraction word problem involving linear equation :

Let x be the given number.
Correct answer is multiplying x with 9⁄16
= x x (9⁄16) = 9x⁄16.

By data, dividing x with 9⁄16
gave 175 more than correct answer which is 9x⁄16.
⇒ x ÷ (9⁄16) = 175 + 9x⁄16
⇒ x x (16⁄9) = 175 + 9x⁄16
⇒ 16x⁄9 = 175 + 9x⁄16
⇒ 16x⁄9 - 9x⁄16 = 175

L.C.M. of the denominators (9, 16) = 9 x 16 = 144

∴ (16 x 16 - 9 x 9)x⁄144 = 175
⇒ 175x⁄144 = 175
⇒ x = 175 x (144⁄175) = 144. Ans.

Solved Example 7 of Fraction Word Problems

A pole has half of its length in the mud, 1⁄3 of its length in water and 1  ²⁄₃ m above the water. Find the whole length of the pole.

Solution to fraction word problem involving linear equation :

Let x m be the length of the pole.
Then, length of the pole in the mud = x x (1⁄2) = x⁄2
Length of the pole in water = x x (1⁄3) = x⁄3
Length of the pole above water = 1  ²⁄₃ = (1 x 3 + 2)⁄3 = 5⁄3

We have, length of the pole =
x = x⁄2 + x⁄3 + 5⁄3
Let us find the L.C.M. of the denominators (2, 3, 3)

        3|    2     3     3
        -------------------
              2     1     1    
        

Solved Example 8 of Fraction Word Problems

A drum of water is 3⁄7 full. When 28 litres are drawn from it, it is just 5⁄14 full. Find the total capacity of the drum, in litres.

Solution to fraction word problem involving linear equation :

Let x litres be the total capacity of the drum.
When the drum is 3⁄7 full,
the volume of water in the drum = x x 3⁄7 = 3x⁄7
When 28 litres are drawn from it,
the volume of water in the drum = 3x⁄7 - 28
By data this is 5⁄14 full.
⇒ volume = x x 5⁄14 = 5x⁄14
∴ 5x⁄14 = 3x⁄7 - 28
⇒ 3x⁄7 - 5x⁄14 = 28
Let us find the L.C.M. of the denominators.

        7|    14     7     
        ----------------
               2     1    
        

Solved Example 9 of Fraction Word Problems

From a class of 44 students, 1⁄5th of the girls and 1⁄8th of the boys took part in a social camp. What fraction of the total strength took part in the camp?

Solution :
Number of girls in the class is a multiple of 5 and
Number of boys in the class is a multiple of 8.

Below 44, Mulitples of 5 = 5, 10, 15, 20, 25, 30, 35, 40.
Below 44, Mulitples of 8 = 8, 16, 24, 32, 40.

To get a sum of 44, the possibilty is 20 girls and 24 boys.

Number of girls that took part in camp = 1⁄5th of 20 = 4.
Number of boys that took part in camp = 1⁄8th of 24 = 3.

So a total of 4 + 3 students out of 44 took part in camp.

Fraction of the total strength took part in the camp = 7⁄44. Ans

Solved Example 10 of Fraction Word Problems

A trian starts with a number of passengers. At the first station, it drops one-third of the passengers and takes 280 more. At the second station, it drops one-half of the new total and takes 12 more. On arriving at the third station, it is found to have 248 passengers. find the number of passengers in the beginning.

Solution to fraction word problem involving linear equation :

let x be the number of passengers in the beginning.

At the first station,
number of passengers dropped = 1⁄3rd of x = x⁄3
number of passengers added = 280.
So, total number of passengers while leaving first platform
= x - x⁄3 + 280. = (3x - x + 280 x 3)⁄3 = (2x + 840)⁄3

At the second station,
number of passengers dropped = 1⁄2 of [(2x + 840)⁄3] = (x + 420)⁄3
number of passengers added = 12.
So, total number of passengers while leaving second platform
= (2x + 840)⁄3 - (x + 420)⁄3 + 12.
= (2x + 840 -x - 420 + 12 x 3)⁄3 = (x + 456)⁄3

By data, this = 248
⇒ (x + 456)⁄3 = 248 ⇒ x + 456 = 3 x 248 = 744
⇒ x = 744 - 456 = 288.

Thus, the number of passengers in the beginning = 288. Ans.

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Exercise on Fraction Word problems

Solve the following Fraction Word problems :

1. 2⁄3 students in a class are boys. If there are 19 girls in the class, how many are boys ?

2. A water tank can hold 56  ¹⁄₄ litres of water. How much water is contained in the tank, when it is 8⁄15 full ?

3. From a string 15 metres long, how many pieces of length 2  ¹⁄₂ metres can be cut ?

4. A packet of toffees weighs 3⁄4 kg. and a tin containing the packets weighs 22  ¹⁄₂ kg. How many packets of toffees arethere in the tin ? (weight of tin ignored.)

5. A person takes 4  ¹⁄₂ minutes to read a page. How many pages can the person read in 36 minutes ?

6. A tumbler holding 1  ¹⁄₂ litres is used to fill a drum which can hold 30 litres. How many times should the water be poured using the tumbler to fill the drum ?

7. Find the fraction which is as much greater than 5⁄8 as is less than 3⁄4.

8. If 1⁄8 of a pencil is black, 1⁄2 of the remaining is white and the remaining 3  ¹⁄₂ cm is blue, find the total length of the pencil.

9. On a certain day, 810 people visited the school exhibition. Out of this, 7⁄15 were men and 11⁄30 were ladies and the rest were children. How many children visited the exhibition?

10. A cake weighs 2 kg. If 2⁄7 of its weight is flour, 1⁄8 of its weight is sugar, 3⁄14 of its weight is milk and the rest is nuts and plums, find the weight of nuts and plums in the cake.

For Answers see at the bottom of the page.

Answers to Exercise on Fraction Word problems

Answers to Fraction Word problems :

1. 38
2. 30 litres
3. 6
4. 30
5. 8
6. 20
7. 11⁄16
8. 8 cm
9. 135
10. 750 g.