UNITARY METHOD WITH FRACTIONS

Example 1 :

If 3/8 of a shipping container holds 2100 identical cartons, how many cartons will fit into :

(a)  5/8 of the cartons 

(b) the whole container

Solution :

Given that  :

3/8 of a shipping container holds 2100 identical cartons.

Let us find the number of identical cartons that 1/8 of shipping container hold.

1/8 of shipping container can hold  =  2100/3

  =  700 cartons

(i)  

Number of identical cartons that 5/8 of shipping container will hold 

=  700(5)

  =  3500 cartons

(ii) 8/8 of container can hold  =  700(8)

  =  5600

Example 2 :

2/11 of Jo’s weekly earnings are paid as income tax. She has $666 remaining after tax. What is her total weekly pay?

Solution :

Let x be Jo's weekly income.

2/11 of his earning are paid as income.

Remaining amount  =  666

x - (2/11) of x  =  666

x-(2x/11)  =  666

9x/11  =  666

x  =  666 ⋅ (11/9)

x  =  814

So, Jo's weekly earning is $814.

Example 3 :

3/13 of a field was searched for truffles and 39 were found. How many truffles would we expect to find in the remainder of the field?

Solution :

Let x be the number of truffles.

3/13 of x  =  39

3x/13  =  39

x  =  39⋅(13/3)

x  =  169

Remaining number of truffles  =  169-39

=  130

Example 4 :

Last week we picked 1/3 of our grapes and this week we picked 1/4 of them. So far we have picked 3682 kg of grapes. What is the total weight of grapes we expect to pick?

Solution :

Let x be total number of grapes.

Number of grapes picked last week =  1/3 of x

=  x/3

Number of grapes picked this week =  (1/4) of x

=  x/4

(x/3) + (x/4)  =  3682

7x/12  =  3682

x  =  3682 ⋅ (12/7)

x  =  6312 kg

So, total weight of grapes is 6312 kg.

Example 5 :

Annika pays 2/25 of her weekly income into a retirement fund. If she pays $42 into the retirement fund, what is her :

a) weekly income

b) annual income?

Solution :

a)  Let x be the Annika's weekly income.

2/25 of x  =  42

2x/25  =  42

x  =  42(25/2)

x  =  525

Her weekly income is $525.

b)  52 weeks  =  1 year

Annual income  =  52(525)

  =  $27300

Example 6 :

Jamil spent 1/4 of his weekly salary on rent, 1/5 on food, and 1/6 on clothing and entertainment. The remaining money was banked.

a) What fraction of Jamil’s money was banked?

b) If he banked $138:00, what is his weekly salary?

c) How much did Jamil spend on food?

Solution :

Let x be his salary.

Money spent for rent  =  1/4 of x

Money spent for food  =  1/5 of x

Money spent for clothing  =  1/6 of x

(a)

Part of money banked  =  x - [(x/4)+(x/5)+(x/6)]

=  x - (37x/60)

=  23x/60

So, the fraction of money banked is 23/60.

(b)  23x/60  =  138

x  =  138(60/23)

x  =  360

His salary is $360.

(c)  Money spent for food  =  x/5

=  360/5

=  72

Example 7 :

In autumn a tree starts to shed its leaves. 2/5 of the leaves fall off in the first week, 1/2 of those remaining fall off in the second week, and 2/3 of those remaining fall off in the third week. 85 leaves now remain.

a) What fraction of leaves have fallen off at the end of:

(i) the second week (ii) the third week?

b) How many leaves did the tree have to start with?

Solution :

Let x be the total number of leaves.

Number of leaves fall off in the first week  =  2/5 of x

=  2x/5

Number of leaves fall off in the second week 

=  (1/2) of 3/5 of x

=  3x/10

Number of leaves fall off in the third week 

=  (2/3) of 3/10 of x

=  x/5

(a) 

(i)  Fraction of leaves fall off at the end of the second week

=  (2x/5) + (3x/10)

=  7x/10

So, the answer is 7/10.

(ii)  Fraction of leaves fall off at the end of the third week

=  (2x/5) + (3x/10) + (x/5)

=  9x/10

So, the answer is 9/10.

(ii)  Total number of leaves initially :

9x/10 + 85  =  x

x/10  =  85

x  =  850

So, the total number of leaves is 850.

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