If 4/7 of an amount of money is $480, then 1/7 of the amount is
= $480 ÷ 4
= $120
Thus 6/7 of the amount = 6x $120 = $720.
and 7/7 is the whole amount which is 7 x $120 = $840.
So, given the value of a number of parts of a quantity we can find one part of the quantity and then the whole quantity or another fraction of the quantity. This is called the unitary method.
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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