Ratio & Proportion Questions and Answers

Answer: B

The philanthropist gave away Rs 2,00,000.

Let there be \(m\) men and \(w\) women, with \(m + w = 300\). The gifts claimed are

\[ \begin{aligned} 750 \cdot \tfrac{8}{9}m + 1000 \cdot \tfrac{2}{3}w &= \tfrac{2000}{3}m + \tfrac{2000}{3}w \\ &= \tfrac{2000}{3}(m + w) \\ &= \tfrac{2000}{3} \times 300 = 200000 \end{aligned} \]

Each claiming man and each claiming woman works out to the same Rs 2000/3 per eligible person, so the split between men and women does not matter. The other amounts come from using only one group's rate for everyone or from slips in the fractions.

Answer: B

Each class has 200 girls.

Let each class have \(g\) girls. Class VIII then has \(450 - g\) boys and class IX \(360 - g\) boys. The condition \(\frac{g}{450 - g} = \frac{360 - g}{g}\) gives

\[ \begin{aligned} g^{2} &= (450 - g)(360 - g) \\ 0 &= 162000 - 810g \\ g &= 200 \end{aligned} \]

Check: VIII has 200 girls and 250 boys (ratio 4:5); IX has 200 girls and 160 boys, so boys to girls is 160:200 = 4:5. 150, 250 and 175 do not make the two ratios equal.

Answer: D

The ratio \(p : u\) is 16 : 45.

Chain the ratios with a common value. Take \(p = 1\); then \(q = 2\). From \(q : r = 4 : 3\), \(r = \frac{3}{4} \times 2 = 1.5\). From \(r : s = 4 : 5\), \(s = \frac{5}{4} \times 1.5 = 1.875\). And \(u = 1.5 \times 1.875 = 2.8125\).

\[ \frac{p}{u} = \frac{1}{2.8125} = \frac{16}{45} \]

The ratio \(p : s\) is 8 : 15; 16 : 15 doubles it instead of scaling \(s\) up by 1.5, and 2 : 15 and 1 : 5 do not follow from chaining the given ratios.

Answer: B

The value is 1.25.

Clear the fractions by multiplying the ratio by 12, the LCM of 2, 3 and 4:

\[ \begin{aligned} x : y : z &= 6 : 4 : 3 \\ \frac{x + z - y}{y} &= \frac{6 + 3 - 4}{4} = \frac{5}{4} = 1.25 \end{aligned} \]

0.75 is \(3/4\), the ratio z : y alone. 2.25 is \((x + z)/y = 9/4\), forgetting to subtract y, and 3.25 is \((x + y + z)/y = 13/4\), adding y instead of subtracting it.

Answer: B

The savings in both years together are ₹ 5,000.

Income is in the ratio 2 : 3, with the second year 45,000, so the first year's income is 30,000. Expenditure is in the ratio 5 : 9, with the first year 25,000, so the second year's is 45,000:

\[ \begin{aligned} \text{Year 1} &: 30000 - 25000 = 5000 \\ \text{Year 2} &: 45000 - 45000 = 0 \end{aligned} \]

Total savings are 5,000. The other options come from misreading which ratio applies to which year.

Answer: B

The new ratio is 22 : 21.

Take 8 and 7 as the original numbers. Increase the boys by 10% and the girls by 20%:

\[ \begin{aligned} 8 \times 1.1 &= 8.8 \\ 7 \times 1.2 &= 8.4 \\ 8.8 : 8.4 &= 22 : 21 \end{aligned} \]

Option A reverses the ratio. 23 : 24 and 24 : 23 come from adding the percentages to the ratio terms instead of multiplying.

Answer: C

Veena paid 3/11 of the bill.

Let Tanya pay \(t\). Veena paid half of that, and Amita two-thirds of Veena's share:

\[ \begin{aligned} V &= t/2,\quad A = \tfrac{2}{3} \cdot \tfrac{t}{2} = \tfrac{t}{3} \\ \text{total} &= t + \tfrac{t}{2} + \tfrac{t}{3} = \tfrac{11t}{6} \end{aligned} \]

Veena's fraction is \((t/2) \div (11t/6) = 3/11\). In whole numbers, Tanya 6, Veena 3, Amita 2, total 11. 1/3 would mean the three paid equally.

Answer: B

The two numbers are in the ratio 4 : 5.

Take the third number as 100. The first is 20% more, 120, and the second 50% more, 150:

\[ 120 : 150 = 4 : 5 \]

2 : 5 compares the percentage increases themselves, which ignores the common base, and 6 : 7 and 3 : 5 do not follow.

Answer: B

The Blue box has 45 balls.

Let the boxes hold \(R\), \(B\) and \(G\) balls. Red and Blue together are twice Green, and Green is twice Red:

\[ \begin{aligned} R + B &= 2G \\ R + B + G &= 90 \\ 3G &= 90,\ G = 30 \end{aligned} \]

\[ \begin{aligned} R &= G/2 = 15 \\ B &= 90 - 30 - 15 = 45 \end{aligned} \]

Check: Red and Blue together hold 60, twice Green's 30, and Green's 30 is twice Red's 15. The option 15 is the Red box and 30 the Green box, so they answer the wrong part of the question.

Answer: A

The tree is 21 m high.

Under similar conditions the sun's rays fall at the same angle, so heights are proportional to the shadows they cast (similar triangles):

\[ \begin{aligned} \frac{h}{15} &= \frac{14}{10} \\ h &= \frac{14 \times 15}{10} = 21\ \text{m} \end{aligned} \]

The shadow is 1.5 times longer, so the tree is 1.5 times taller than the pole. 14 m is the pole itself, 15 m merely repeats the shadow length, and 25 m comes from adding the 10 m difference instead of scaling.