Ratio & Proportion Questions and Answers

Answer: D

The value is 39 : 41.

Put \(x = 9k\) and \(y = 11k\); the \(k\) cancels:

\[ \begin{aligned} \frac{5(9) + 3(11)}{3(9) + 5(11)} &= \frac{45 + 33}{27 + 55} \\ &= \frac{78}{82} = \frac{39}{41} \end{aligned} \]

45 : 55 and 18 : 22 are just multiples of 9 : 11, as if the expression did not change the ratio, and 37 : 41 is an arithmetic slip.

The case of the letters (X : Y in the ratio, x and y in the expression) is a printing inconsistency; both refer to the same two quantities.

Answer: A

The whole property is worth ₹ 16,250.

Son : wife = 3 : 1 and wife : daughter = 3 : 1, so son : wife : daughter = 9 : 3 : 1.

\[ \begin{aligned} 9k - k &= 8k = 10000 \Rightarrow k = 1250 \\ \text{total} &= 13k = 16250 \end{aligned} \]

Treating the ratio as son : wife : daughter = 3 : 1 : 1/3 leads to the same answer; the other totals do not split 9 : 3 : 1 with a difference of 10,000.

Answer: B

The sum is 8.

The ratio has \(1 + 2 + 3 + 4 = 10\) parts, so each part is \(\frac{16}{10} = 1.6\). The first and fourth numbers together are \(1 + 4 = 5\) parts:

\[ 5 \times 1.6 = 8 \]

The numbers are 1.6, 3.2, 4.8 and 6.4. 5 is the number of parts, not their value, and 80 forgets to divide by 10.

Shortcut: the first and fourth parts are half of all the parts (5 of 10), so their sum is half of 16.

Answer: C

The correct matching is a-III, b-II, c-I, d-IV.

The third proportional to \(a\) and \(b\) is \(b^2/a\); the fourth proportional to \(a, b, c\) is \(bc/a\).

\[ \begin{aligned} \text{a: } 32^2/16 &= 64 \\ \text{b: } 18^2/6 &= 54 \\ \text{c: } 7 \times 9/3 &= 21 \\ \text{d: } 16 \times 7/4 &= 28 \end{aligned} \]

So a goes with 64 (III), b with 54 (II), c with 21 (I) and d with 28 (IV). Option A swaps b and c, giving 6 and 18 a third proportional of 21, which does not satisfy \(6 : 18 = 18 : x\).

Answer: B

The value of the gift is ₹ 140, reading the stem as the key does: Sonu is left with two thirds of his amount.

Let the savings be \(4x\) and \(5x\) and the gift cost \(c\), shared 3:4, so Sonu pays \(\tfrac{3c}{7}\) and Monu \(\tfrac{4c}{7}\). Sonu keeps two thirds of \(4x\), so his share is one third of it:

\[ \begin{aligned} \tfrac{3c}{7} &= \tfrac{4x}{3} \\ c &= \tfrac{28x}{9} \end{aligned} \]

Monu is left with 145:

\[ \begin{aligned} 5x - \tfrac{4}{7}\cdot\tfrac{28x}{9} &= 145 \\ 5x - \tfrac{16x}{9} &= 145 \\ \tfrac{29x}{9} &= 145 \\ x &= 45 \end{aligned} \]

\[ c = \tfrac{28 \times 45}{9} = 140 \]

Check: Sonu has 180 and pays 60, keeping 120, which is two thirds of 180; Monu has 225 and pays 80, keeping 145. The stem as printed says Sonu spent two thirds, which read literally gives no whole-number answer among the options (the share would be 120 and the data inconsistent), so the key only works with the left-with reading. 165, 180 and 185 do not fit either reading.

Answer: A

Quantity of milk \(= (60 \times \frac{2}{3}) littres = 40 littres.\)

Quantity of water in it \(= (60 - 40) littres = 20 littres.\)

New ratio \(= 1:2\)

Let quantity of water to be added further be \(x \) \(littres.\)

Then, milk : water \(= (\frac{40}{20 + x})\)

Now, \((\frac{40}{20 + x}) = \frac{1}{2}\)

\(\Rightarrow 20 + x = 80\)

\(\Rightarrow x = 60.\)

\(\therefore\) Quantity of water to be added \(= 60\)  \(littres.\)

Answer: C

Let the number of \(25 p, 10 p\) and \(5 p\) coins be \(x, 2x, 3x \) respectively.

Then, sum of there value \(= Rs. (\frac{25x}{100} + \frac{10 \times 2x}{100} + \frac{5 \times 3x}{100}) = Rs. \frac{60x}{100}\)

\(\therefore \frac{60x}{100} = 30 \Leftrightarrow x = \frac{300 \times 100}{60} = 50.\) 

Heance, the number of \(5 p\) coins \(= (3 \times 50) = 150.\)

Answer: C

Let \(40\)% of \(A = \frac{2}{3} B\)

Then, \(\frac{40A}{100} = \frac{2B}{3}\)

\(\Rightarrow \frac{2A}{5} = \frac{2B}{3}\)

\(\Rightarrow \frac{A}{B} = (\frac{2}{3} \times \frac{5}{2}) = \frac{5}{3}\)

\(\therefore A : B = 5 : 3.\)

Answer: C

The proportion means \( \tfrac{0.75}{x} = \tfrac{5}{8} \).

\[ 5x = 0.75 \times 8 = 6 \]

\[ x = 1.2 \]

Answer: C

\(\frac{4}{12}A = \frac{2}{5}B\)

\(\Rightarrow A = (\frac{2}{5} \times \frac{15}{4})B\)

\(\Rightarrow A = \frac{3}{2} B\)

\(\Rightarrow \frac{A}{B} = \frac{3}{2}\)

\(\Rightarrow A : B = 3 : 2.\)

\(\therefore B's\) share \(= RS. (1210 \times \frac{2}{5}) = Rs. 484\)