Ratio & Proportion Questions and Answers

Answer: A

The ratio is 11 : 9.

Add and subtract the two equations:

\[ \begin{aligned} a + b &= 40,\ a - b = 4 \\ 2a &= 44,\ a = 22,\ b = 18 \\ 22 : 18 &= 11 : 9 \end{aligned} \]

11 : 18 mixes the reduced first number with the unreduced second, and 21 : 19 and 7 : 6 come from wrong values for the numbers. Only 11 : 9 gives numbers that add to 40 and differ by 4.

Answer: B

The total strength is 896.

Boys and girls are in the ratio 9 : 5, so the school has 9 + 5 = 14 parts, of which the girls are 5.

\[ \begin{aligned} 5 \text{ parts} &= 320 \\ 1 \text{ part} &= 64 \\ 14 \text{ parts} &= 14 \times 64 = 896 \end{aligned} \]

There are 576 boys, which is the trap option: it is the number of boys, not the total. 840 and 920 do not divide into 14 equal parts of 64.

Answer: B

The ratio of the three pieces is 33 : 28 : 49.

Take the whole pizza as 10 parts: the pieces are 3 and 7. The bigger piece, 7, is cut in the ratio 4 : 7, that is into 4/11 and 7/11 of itself.

\[ \begin{aligned} 7 \times \tfrac{4}{11} &= \tfrac{28}{11}, \quad 7 \times \tfrac{7}{11} = \tfrac{49}{11} \\ 3 &= \tfrac{33}{11} \end{aligned} \]

Multiplying all three by 11 gives 33 : 28 : 49 (total 110, which checks with 3 : 7 since 33 : 77). The other ratios do not keep the first cut at 3 : 7.

Answer: B

The larger pulley revolves at 64 rpm.

Speed is inversely proportional to diameter, so speed times diameter is the same for both pulleys:

\[ \begin{aligned} 12 \times N &= 8 \times 96 \\ N &= \frac{768}{12} = 64\ \text{rpm} \end{aligned} \]

The larger pulley must turn more slowly. 144 rpm is what you get by treating the relation as direct proportion, \(96 \times \frac{12}{8}\), and 40 and 48 rpm do not satisfy \(12N = 768\).

Answer: A

The ratio a:c is 3:7.

Make the b terms equal. In the first ratio b is 9 and in the second 27, so multiply the first ratio by 3:

\[ \begin{aligned} a : b &= 5 : 9 = 15 : 27 \\ b : c &= 27 : 35 \\ a : c &= 15 : 35 = 3 : 7 \end{aligned} \]

7:3 inverts the ratio, and 4:7 and 7:4 do not follow from the given ratios.

Answer: C

The ratio is 2:3.

Let the first number be \(a\) and the second \(b\). Adding half of \(a\) to \(b\) makes \(b\) four-thirds of itself.

\[ \begin{aligned} b + \frac{a}{2} &= \frac{4b}{3} \\ \frac{a}{2} &= \frac{b}{3} \\ \frac{a}{b} &= \frac{2}{3} \end{aligned} \]

Check with \(a = 2\), \(b = 3\): \(3 + 1 = 4\), which is \(\frac{4}{3}\) of 3. 3:2 is the same ratio inverted, the usual slip.

Answer: B

The number of boys is 265.

Girls : boys = 4 : 5, and 212 girls correspond to 4 parts, so one part is 53:

\[ \text{boys} = \frac{212}{4} \times 5 = 53 \times 5 = 265 \]

256, 251 and 270 do not keep the ratio 4 : 5 with 212 girls; 256 merely swaps the last two digits of the answer.

Check: dividing both 212 and 265 by 53 gives 4 and 5, so the ratio is exactly 4 : 5.

Answer: D

The ratio is 3:1.

Since \(x : y = 3 : 4\), put \(x = 3k\) and \(y = 4k\).

\[ \begin{aligned} 2x + 3y &= 6k + 12k = 18k \\ 3y - 2x &= 12k - 6k = 6k \\ 18k : 6k &= 3 : 1 \end{aligned} \]

1:3 is the ratio turned upside down, and 3:2 and 2:3 come from arithmetic slips in one of the two brackets.

Answer: D

The numbers are (18, 42).

Let a = 3k and b = 7k. After adding 9 to each:

\[ \begin{aligned} \frac{3k + 9}{7k + 9} &= \frac{9}{17} \\ 51k + 153 &= 63k + 81 \\ 12k &= 72, \quad k = 6 \end{aligned} \]

So a = 18 and b = 42. Check: 27/51 = 9/17. (9, 21) and (15, 35) are also in the ratio 3:7 but give 18/30 and 24/44, and (8, 42) is not in the ratio 3:7 at all.

Answer: B

A's income is ₹ 25,000.

Let the incomes be 5x and 6x and the expenses 4y and 5y. Each saves ₹ 5,000, so:

\[ \begin{aligned} 5x - 4y &= 5000 \\ 6x - 5y &= 5000 \end{aligned} \]

Multiply the first equation by 5 and the second by 4, then subtract to remove y:

\[ \begin{aligned} 25x - 20y &= 25000 \\ 24x - 20y &= 20000 \\ x &= 5000 \end{aligned} \]

So A's income is 5x = ₹ 25,000. Check: y = 5,000, so A spends ₹ 20,000 and saves ₹ 5,000; B earns ₹ 30,000, spends ₹ 25,000 and saves ₹ 5,000. ₹ 30,000 is B's income, not A's.