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: D

The number of girls is 30.

Boys : girls = 4 : 5, and the 4 parts are 24 boys, so one part is 6:

\[ \begin{aligned} 4k &= 24 \Rightarrow k = 6 \\ \text{girls} &= 5k = 30 \end{aligned} \]

Check: 24 : 30 reduces to 4 : 5. Since girls outnumber boys in this ratio, any answer below 24 (20, 19 or 16) is impossible; 20 is the result of applying the ratio the wrong way round.

Answer: C

The speed ratios are Elephant : Lion = 20 : 80 = 1 : 4, Lion : Cheetah = 80 : 100 = 4 : 5, Elephant : Cheetah = 20 : 100 = 1 : 5, Lion : Elephant = 4 : 1.

The least ratio is 1 : 5, that of the Elephant and the Cheetah, option C.

Ratios are compared as fractions: 1/4 = 0.25 for the Elephant and the Lion, 4/5 = 0.8 for the Lion and the Cheetah, and 1/5 = 0.2 for the Elephant and the Cheetah. The smallest value is 0.2, so the Elephant to Cheetah ratio is the least. The Lion to Elephant ratio is 4, the largest.

Answer: C

The ratio 7 : 17.5 equals 2 : 5. From x : 7.5 = 2 : 5 we get \[x=\tfrac{2\times 7.5}{5}=3,\] and from 8 : y = 2 : 5 we get \[y=\tfrac{8\times 5}{2}=20.\]

So x : y = 3 : 20, option C. The options 3 : 22 and 3 : 23 come from a wrong y, and 1 : 7 does not follow from either ratio.

A quick check: x = 3 gives 3 : 7.5 = 2 : 5 and y = 20 gives 8 : 20 = 2 : 5, so all three ratios are equal to 2 : 5, as the question requires.

Answer: B

The total of 5 + 3 = 8 parts is Rs 1,600, so one part is Rs 200. \[Q\text{'s share}=3\times 200=600.\]

Rs 1,000 would be P's share if P had 5 parts, and 900 or 1,200 do not match the 3-part share, so the answer is 600, option B.

As a check, P gets 5 x 200 = Rs 1,000 and Q gets Rs 600, which adds to Rs 1,600 in the ratio 5 : 3.

Answer: B

The smaller number is 126.

Write the numbers as \(7k\) and \(9k\). Their difference is \(2k\):

\[ \begin{aligned} 9k - 7k &= 36 \\ k &= 18 \\ 7k &= 126,\ 9k = 162 \end{aligned} \]

Check: \(162 - 126 = 36\) and \(126 : 162 = 7 : 9\). 112 is \(7 \times 16\), which would need a difference of 32, 168 is \(7 \times 24\), which would need 48, and 144 is not a multiple of 7 at all.