The chapter on Proportion in 6th-grade mathematics introduces the concept of comparing two ratios. When two ratios are equal, the four quantities involved are said to be in proportion.
1. Definition of Proportion
Two ratios a:b and c:d are in proportion if:
We write this as a : b :: c : d, which is read as “a is to b as c is to d.”
- Extremes: The first and fourth terms (a and d) are called the extremes.
- Means: The second and third terms (b and c) are called the means.
2. The Fundamental Property
For any four quantities to be in proportion, the product of the means must equal the product of the extremes:
{Product of Means} (b * c) = {Product of Extremes} (a * d)
If this equality holds, the numbers are in proportion.
3. Continued Proportion
Three quantities a, b, and c are said to be in continued proportion if:
a : b :: b : c
In this case, b is known as the mean proportional between a and c. The relationship is expressed as b^2 = ac.
4. Unitary Method
Proportion is often used to solve real-world problems using the Unitary Method:
- Find the value of one unit: Divide the total value by the number of items.
- Find the value of the required number of units: Multiply the value of one unit by the required quantity.
Exercise 12 (A)
Question 1.
In each of the following, check whether or not the given ratios form a proportion :
(i) 8 : 16 and 12 : 15
(ii) 16 : 28 and 24 : 42
(iii) 12 ÷ 3 and 8 ÷ 2
(iv) 25 : 40 and 20 : 32
Solution :
Question 2.
Find the value of x in .each of the following proportions :
(i) x : 4 = 6 : 8
(ii) 14 : x = 7 : 9
(iii) 4 : 6 = x : 18
(iv) 8 : 10 = x : 25
(v) 5 : 15 = 4 : x
(vi) 16 : 24 = 6 : x
Solution :
Question 3.
Find the value of x so that the given four numbers are in proportion :
(i) x, 6, 10 and 15
(ii) x, 4, 15 and 30
(iii) 2, x, 10 and 25
(iv) 4, x, 6 and 18
(v) 9, 12, x and 8
(vi) 4, 10, 36 and x
(vii) 7, 21, x and 45
(viii) 6, 8, 12 and x.
Solution :
Question 4.
The first, second and the fourth terms of a proportion are 6, 18 and 75, respectively. Find its third term.
Solution :
Question 5.
Find the second term of the proportion whose first, third and fourth terms are 9, 8 and 24 respectively.
Solution :
Question 6.
Find the fourth term of the proportion whose first, second and third terms are 18, 27, and 32 respectively.
Solution :
Question 7.
The ratio of the length and the width of a school ground is 5 : 2. Find the length, if the width is 40 metres.
Solution :
Question 8.
The ratio of the sale of eggs on a Sunday and that of the whole week at a grocery shop
was 2 : 9. If the total value of the sale of eggs in the same week was Rs 360, find the
value of the sale of eggs that Sunday.
Solution :
Question 9.
The ratio of copper and zinc in an alloy is 9 : 8. If the weight of zinc, in the alloy, is 9.6 kg ; find the weight of copper in the alloy.
Solution :
Question 10.
The ratio of the number of girls to the number of boys in a school is 2 : 5. If the number of boys is 225 ; find:
(i) the number of girls in the school.
(ii) the number of students in the school.
Solution :
Question 11.
In a class, one out of every 5 students pass. If there are 225 students in all the sections of a class, find how many pass ?
Solution :
Question 12.
Make set of all possible proportions from the numbers 15, 18, 35 and 42.
Solution :
EXERCISE 12(B)
Question 1.
If x, y and z are in continued proportion, then which of the following is true :
(i) x : y = x : z
(ii) x : x = z : y
(iii) x : y = y : z
(iv) y : x = y : z
Solution :
(iii) x : y = y : z
Question 2.
Which of the following numbers are in continued proportion :
(i) 3, 6 and 15
(ii) 15, 45 and 48
(iii) 6, 12 and 24
(iv) 12, 18 and 27
Solution :
(iii) and (iv)
Question 3.
Find the mean proportion between
(i) 3 and 27
(ii) 0.06 and 0.96
Solution :
Question 4.
Find the third proportional to :
(i) 36, 18
(ii) 5.25, 7
(iii) ₹ 1.60, ₹ 0.40
Solution :
Question 5.
The ratio between 7 and 5 is same as the ratio between ₹ x and ₹ 20.50 ; find the value of x.
Solution :
Since, It is given that the ratio between 7 and 5 is same as the ratio between ₹ x and ₹ 20.50
Question 6.
If (4x + 3y) : (3x + 5y) = 6 : 7, find :
(i) x : y
(ii) x, if y = 10
(iii) y, if x = 27
Solution :
Question 7.
(i) x : y
(ii) x, if y = 70
(iii) y, if x = 33
Solution :
EXERCISE 12(C)
Question 1.
Are the following numbers in proportion:
(i) 32, 40, 48 and 60 ?
(ii) 12,15,18 and 20 ?
Solution :
which is not true.
12, 15, 18 and 20 are not in proportion.
Question 2.
Find the value of x in each of the following such that the given numbers are in proportion.
(i) 14, 42, x and 75
(ii) 45, 135, 90 and x
Solution :
Question 3.
The costs of two articles are in the ratio 7 : 4. If the cost of the first article is Rs. 2,800 ; find the cost of the second article.
Solution :
Ratio in the cost of two articles = 7 : 4
Cost of first article = Rs. 2800
Let cost of the second article = x
7 : 4 = 2800 : x
Question 4.
The ratio of the length and the width of a rectangular sheet of paper is 8 : 5. If the width of the sheet is 17.5 cm; find the length.
Solution :
Let length of sheet = x cm
Ratio in length and breadth = 8 : 5
and width = 17.5 cm
8 : 5 = x : 17.5
Question 5.
The ages of A and B are in the ratio 6 : 5. If A’s age is 18 years, find the age of B.
Solution :
Ratio in the ages of A and B = 6 : 5
A’s age = 18 years
Let B’s age = x years
6 : 5 = 18 : x
Question 6.
A sum of Rs. 10, 500 is divided among A, B and C in the ratio 5 : 6 : 4. Find the share of each.
Solution :
Total amount = Rs. 10, 500
Ratio in A, B, and C = 5 : 6 : 4
Sum of ratio = 5 + 6 + 4 = 15
Question 7.
Do the ratios 15 cm to 2 m and 10 sec to 3 minutes form a proportion?
Solution :
15 cm : 2 m : : 10 sec : 3 min
15 cm : 2 x 100 cm :: 10 sec : 30 x 60 sec
15 : 200 :: 10 : 1800
3 : 40 :: 1 : 180
No, they donot form a proportion
Question 8.
Do the ratios 2 kg : 80 kg and 25 g : 625 g form a proportion ?
Solution :
2 kg : 80 kg : : 25 g : 625 g
2 : 80 :: 25 : 625
1 : 40 :: 1 : 25
No, they do not form a proportion.
Question 9.
10 kg sugar cost ₹ 350. If x kg sugar of the same kind costs ₹ 175, find the value of x
Solution :
Question 10.
The length of two ropes are in the ratio 7 : 5. Find the length of:
(i) shorter rope, if the longer one is 22.5 ni
(ii) longer rope, if the shorter is 9.8 m.
Solution :
Length of the ropes are in the ratio = 7 : 5
(i) Let the length of shorter rope = x
Length of longer rope = 22.5 m
A.T.Q.
Question 11.
If 4, x and 9 are in continued proportion, find the value of x.
Solution :
4, x and 9 are in continued proportion
=> 4 : x = x : 9
=> x2 = 9 x 4
=> x = √36
x = 6
Question 12.
If 25, 35 and x are in continued proportion, find the value of x.
Solution :
25, 35 and x are in continued proportion
=> 25 : 35 = 35 : x
=> 25 x x = 35 x 35
