Ratio and Proportion 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 $\inline \sqrt{ab}$

4. (i) COMPARISON OF RATIOS : We say that (a: b) > (c: d) $\inline \fn_jvn \Leftrightarrow$ (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 $\inline (a^2:b^2)$.

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

(iii)Triplicate ratio of (a : b) is $\inline (a^3:b^3)$.

(iv) Sub-triplicate ratio of (a : b) is $\inline (a^\frac{1}{3}:b^\frac{1}{3})$.

(V) $\inline \fn_phv If \frac{a}{b}=\frac{c}{d},then\frac{a+b}{a-b}=\frac{c+d}{c-d}$ (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, $\inline \fn_phv x\propto y$

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

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

287 41281
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

$\inline \fn_jvn \frac{4x}{(3x+5)}=\frac{4}{5}$

20x = 4(3x+5)

8x = 20

x = 2.5

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

36 15313
Q:

If Rs. 782 be divided into three parts, proportional to $\inline \frac{1}{2}:\frac{2}{3}:\frac{3}{4}$, then the first part is?

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

Explanation:

Given ratio =$\inline \frac{1}{2}:\frac{2}{3}:\frac{3}{4}$= 6:8:9

$\inline \therefore$1st part = $\inline Rs.[782\times \frac{6}{23}]$ = Rs. 204

50 9484
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

28 9410
Q:

Jo's collection contains US, Indian and British stamps. If the ratio of US to Indian stamps is 5 to 2 and the ratio of Indian to British stamps is 5 to 1, what is the ratio of US to British stamps?

 A) 10 : 5 B) 15 : 2 C) 20 : 2 D) 25 : 2

Explanation:

Indian stamps are common to both ratios. Multiply both ratios by factors such that the Indian stamps are represented by the same number.

US : Indian = 5 : 2, and Indian : British = 5 : 1. Multiply the first by 5, and the second by 2.

Now US : Indian = 25 : 10, and Indian : British = 10 : 2

Hence the two ratios can be combined and US : British = 25 : 2