# Ratios and Proportions Questions

FACTS  AND  FORMULAE  FOR  RATIO  AND  PROPORTION  QUESTIONS

1. RATIO: The ratio of two quantities a and b in the same units, is the fraction a/b and we write it as a:b.

In the ratio a:b, we call a as the first term or antecedent and b, the second term or consequent.

Ex. The ratio 5: 9 represents 5/9 with antecedent = 5, consequent = 9.

Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio. Ex. 4: 5 = 8: 10 = 12: 15 etc. Also, 4: 6 = 2: 3.

2. PROPORTION : The equality of two ratios is called proportion

If a: b = c: d, we write, a: b :: c : d and we say that a, b, c, d are in proportion . Here a and d are called extremes, while b and c are called mean terms.

Product of means = Product of extremes.

Thus, a: b :: c : d <=> (b x c) = (a x d).

3(i) Fourth Proportional : If a : b = c: d, then d is called the fourth proportional to a, b, c.

(ii) Third Proportional : If a: b = b: c, then c is called the third proportional to a and b.

(iii) Mean Proportional : Mean proportional between a and b is $\sqrt{ab}$

4. (i) COMPARISON OF RATIOS : We say that (a: b) > (c: d) $⇔$ (a/b)>(c /d).

(ii) COMPOUNDED RATIO : The compounded ratio of the ratios (a: b), (c: d), (e : f)  is    (ace: bdf)

5. (i) Duplicate ratio of (a : b) is $\left({a}^{2}:{b}^{2}\right)$.

(ii) Sub-duplicate ratio of (a : b) is (√a : √b).

(iii)Triplicate ratio of (a : b) is $\left({a}^{3}:{b}^{3}\right)$.

(iv) Sub-triplicate ratio of (a : b) is .

(V)  (Componendo and dividendo)

6. VARIATION:

(i) We say that x is directly proportional to y, if x = ky for some constant k and we write, $x\propto y$

(ii) We say that x is inversely proportional to y, if xy = k for some constant k and we write, $x\propto \frac{1}{y}$

Q:

A bag contains 50 P, 25 P and 10 P coins in the ratio 5: 9: 4, amounting to Rs. 206. Find the number of coins of each type respectively.

 A) 360, 160, 200 B) 160, 360, 200 C) 200, 360,160 D) 200,160,300

Explanation:

let ratio be x.

Hence no. of coins be 5x ,9x , 4x respectively

Now given total amount = Rs.206

=> (.50)(5x) + (.25)(9x) + (.10)(4x) = 206

we get x = 40

=> No. of 50p coins = 200

=> No. of 25p coins = 360

=> No. of 10p coins = 160

760 120958
Q:

A mixture contains alcohol and water in the ratio 4 : 3. If 5 liters of water is added to the mixture, the ratio becomes 4: 5. Find the quantity of alcohol in the given mixture.

 A) 10 B) 12 C) 15 D) 18

Explanation:

Let the quantity of alcohol and water be 4x litres and 3x litres respectively

4x/(3x+5) = 4/5

20x = 4(3x+5)

8x = 20

x = 2.5

Quantity of alcohol = (4 x 2.5) litres = 10 litres.

160 40900
Q:

Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:

 A) 2:5 B) 3:5 C) 4:5 D) 5:4

Explanation:

Let the third number be x.

Then, first number = 120% of x =120x/100 = 6x/5

Second number =150% of x = 150x/100 = 3x/2

Ratio of first two numbers = 6x/5 : 3x/2 = 12x : 15x = 4 : 5

87 37612
Q:

Salaries of Ravi and Sumit are in the ratio 2:3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40:57. What is Sumit's salary?

 A) 38000 B) 46800 C) 36700 D) 50000

Explanation:

Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.
Then,
(2x+4000) / (3x+4000) = 40 / 57
⇒ 57 × (2x + 4000) = 40 × (3x+4000)
⇒ 6x = 68,000
⇒ 3x = 34,000
Sumit's present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000

104 33697
Q:

A sum of Rs.312 was divided among 100 boys and girls in such a way that the boy gets Rs.3.60 and each girl Rs. 2.40 the number of girls is

 A) 35 B) 40 C) 45 D) 50

Explanation:

Step (i): Let x be the number of boys and y be the number of girls.
Given total number of boys and girls = 100
x+y=100 -------------- (i)

Step (ii): A boy gets Rs. 3.60 and a girl gets Rs. 2.40
The amount given to 100 boys and girls = Rs. 312
3.60x + 2.40y = 312 -------------- (ii)

Step (iii):
Solving (i) and (ii)
3.60x + 3.60y = 360 --------- Multiply (i) by 3.60
3.60x + 2.40y = 312 --------- (ii)
1.20y = 48
y = 48 / 1.20
= 40

$\inline \fn_jvn \Rightarrow$ Number of girls = 40

136 32864
Q:

If Rs. 782 be divided into three parts, proportional to $12:23:34$, then the first part is?

 A) Rs. 182 B) Rs. 190 C) Rs. 192 D) Rs. 204

Explanation:

$Rs.782*623$

Given ratio =$12:23:34$= 6:8:9

1st part = $Rs.782*623$ = Rs. 204

155 29696
Q:

If 0.75: x :: 5:8, then x is equal to:

 A) 1.12 B) 1.16 C) 1.20 D) 1.30

Explanation:

(x * 5) = (0.75 *8)

X=6/5 = 1.20

84 27700
Q:

The compounded ratio of (2 : 3), (6: 11) and (11 :2) is

 A) 1:2 B) 2:1 C) 11:24 D) 36:121

Explanation:

Compounded Ratio :: When we compound two or more ratio's with each other through product or multiplication, the result is simply a compound ratio.

Thus, the product of two or more ratios; i.e, ab:cd is a ratio compounded of the simple ratios a:c and b:d.

Required compounded ratio = (2/3 x 6/11 x 11/2) = 2/1.