Ratio and Proportion

Q1: Fill in the blanks:

i. In a : b, a is called the ______ while b is called the _____.

Answer: antecedent; consequent

ii. a : b > c : d if _____ > _____.

Answer: a × d > b × c

iii. The simplest ratio of 3 hours : 1 day is _____.

1 day = 24 hours
So, 3 hours : 24 hours = 3 : 24 = 1 : 8
Answer: 1 : 8

iv. If 10% of x is equal to 20% of y, then x : y is equal to _____.

\[ \frac{10}{100}x = \frac{20}{100}y \Rightarrow x : y = 2 : 1 \] Answer: 2 : 1

v. The mean proportion between 49 and 81 is _____.

\[ \sqrt{49 \times 81} = \sqrt{3969} = 63 \] Answer: 63

vi. The ratio \(\frac{1}{2} : \frac{1}{3} : \frac{1}{4}\) is the same as _____.

Take LCM of denominators = 12
Multiply each term by 12: \[ \frac{1}{2} \times 12 = 6,\quad \frac{1}{3} \times 12 = 4,\quad \frac{1}{4} \times 12 = 3 \\ \Rightarrow 6 : 4 : 3 \] Answer: 6 : 4 : 3

vii. If a sum of ₹455 is divided between A and B in the ratio 4 : 3, then A gets _____.

Total parts = 4 + 3 = 7
A’s share = \[ \frac{4}{7} \times 455 = ₹260 \] Answer: ₹260

Q2: Write true (T) or false (F):

i. \(a : b :: c : d \Leftrightarrow ac = bd\)

Let’s cross-multiply the proportion: \[ a : b :: c : d \Rightarrow \frac{a}{b} = \frac{c}{d} \Rightarrow ad = bc \ (\text{Not } ac = bd) \] Answer: False (F)

ii. \(\frac{1}{x} : \frac{1}{y}\) is the same as \(y : x\).

\[ \frac{1}{x} : \frac{1}{y} = \frac{1/x}{1/y} = \frac{y}{x} = y : x \] Answer: True (T)

iii. Three numbers a, b, c are in continued proportion if \(\frac{a}{b} = \frac{c}{b}\)

Incorrect.
In continued proportion: \[ \frac{a}{b} = \frac{b}{c} \\ \Rightarrow b^2 = ac \] Answer: False (F)

iv. If \(a : b :: b : c\), then b is called the mean proportion between a and c.

Correct. This is the definition of mean proportion:
If \(\frac{a}{b} = \frac{b}{c}\), then \(b^2 = ac\) ⇒ b is mean proportional.
Answer: True (T)