Ratio & Proportion Questions and Answers

Answer: C

The value of x is 9000. The ratio of A to C is 3 : 7 and of C to B is 4 : 5. Making C the same in both, A : C = 12 : 28 and C : B = 28 : 35, so A : C : B = 12 : 28 : 35 with 75 parts in all.

\[ 28 - 12 = 16 \text{ parts} = 1920 \]

\[ 1 \text{ part} = 120 \]

\[ x = 75 \times 120 = 9000 \]

The options 8400, 8640 and 9120 do not equal 75 parts of 120, so the third option is correct. As a check, one part is 120, so A has 1440, C has 3360 and B has 4200. These add up to 9000, and A and C differ by 1920 as the question says.

Answer: C

The change is 33.33% (a decrease).

Since x = k y / z², raising y and z by 50% multiplies y by 1.5 and z² by 2.25.

\[ \frac{1.5}{2.25}=\frac{2}{3} \]

So x becomes two-thirds of its old value, a fall of 33.33%. 15%, 30% and 35% do not match this factor.

When one quantity is directly proportional to y and inversely proportional to the square of z, the new value is found by multiplying the old value by the changes. A rise of 50% is a factor 1.5. Dividing by 1.5 squared, 2.25, and multiplying by 1.5 gives the factor 2/3.

Answer: B

The third proportion is 1.

The third proportional to a and b is b²/a. Here a = a² + 2a + 1 = (a + 1)² and b = a + 1.

\[ \frac{(a+1)^{2}}{(a+1)^{2}}=1 \]

(a − 1) and the two fractions with (a − 1) do not equal this value.

The third proportion c of a and b satisfies a : b = b : c, so c = b² / a. Since a is a perfect square, (a + 1)², the fraction cancels completely and gives exactly 1. This is the usual shortcut for such problems.

Answer: A

15 apples cost ₹600.

6 apples cost ₹240, so one apple costs ₹40.

\[ 15\times40=600 \]

₹800, ₹640 and ₹840 would need a price per apple that is not 40.

This is a direct proportion: the cost goes up in the same ratio as the number of apples. The number goes from 6 to 15, a factor of 2.5, and 240 × 2.5 = 600. The same result comes from the unit price.

Answer: B

The constant is 1/21.

Z varies jointly as X and Y, so Z = kXY. Substituting Z = 34, X = 14 and Y = 51 gives 34 = k × 714.

\[ k=\frac{34}{14\times51}=\frac{34}{714}=\frac{1}{21} \]

The other fractions would give a different value of Z.

The constant of proportionality is the number that multiplies X and Y to give Z. Putting the values back checks it: (1/21) × 14 × 51 = 714/21 = 34, as given. The other fractions would give 170, 68 or 272/3 rather than 34.

Answer: B

The ratio is 11 : 10.

If girls are 100 parts, boys are 10% more, which is 110 parts.

\[ \text{boys : girls}=110:100=11:10 \]

10 : 11 is the reverse, and 11 : 9 and 9 : 10 do not match a 10% excess.

An easy check is to choose 100 girls. Boys are 10% more, which is 110. The ratio 110 : 100 reduces to 11 : 10, with boys first, as the question asks. The reverse ratio 10 : 11 would put girls first.

Answer: C

The ratio is 21 : 22. Let the numbers be \(10k\) and \(13k\). \[ \frac{10k-90}{13k-90} = \frac{7}{10} \] \[ 100k - 900 = 91k - 630 \] \[ 9k = 270,\quad k = 30 \] So A = 300 and B = 390. Adding 57 to A and subtracting 16 from B: \[ A = 357,\quad B = 374 \] \[ 357 : 374 = 21 : 22 \] since both are divisible by 17. The ratios 19 : 20, 20 : 21 and 22 : 23 do not match 357 : 374. Check: with \(k = 30\) the numbers are 300 and 390, and after subtracting 90 they become 210 and 300, which is 7 : 10 as the question says.

Answer: A

The value of \(x\) is 24. Direct variation with \(y\) and inverse variation with \(z\) means \(x = \frac{ky}{z}\). From the first case: \[ \begin{aligned} 5 &= \frac{k\times 1.5}{2} \Rightarrow k \\ &= \frac{20}{3} \end{aligned} \] For the second case: \[ \begin{aligned} x &= \frac{20}{3}\times\frac{9}{2.5} \\ &= \frac{20\times 9}{3\times 2.5} \\ &= 24 \end{aligned} \] Because \(x\) falls as \(z\) rises and rises with \(y\), the value must exceed 5 by the factor \(\frac{9}{1.5}\times\frac{2}{2.5} = 4.8\), and \(5\times 4.8 = 24\); 20, 15 and 12 do not satisfy this.

Answer: A

The cement costs 400 rupees.

Take 3 parts of sand and 1 part of cement. If one part of sand costs 1 unit, one part of cement costs 2 units, so the costs are

\[ \begin{aligned} \text{sand} &= 3 \times 1 = 3 \\ \text{cement} &= 1 \times 2 = 2 \end{aligned} \]

The cost ratio is therefore 3 : 2, and cement takes 2/5 of the total: 2/5 × 1000 = 400 rupees.

600 is the sand’s share, not the cement’s. 200 is the value of a single cost unit, forgetting that the cement accounts for two of the five. 800 cannot be right, since it would leave only 200 rupees for three times as much sand at half the unit price.

Answer: D

The ratio \(\frac{x}{y}\) is a constant.

Proportionality means \(x + y = k(x - y)\) for a fixed constant \(k\). Collect the terms:

\[ \begin{aligned} x + y &= kx - ky \\ y(1 + k) &= x(k - 1) \\ \frac{x}{y} &= \frac{k + 1}{k - 1} \end{aligned} \]

The right-hand side contains only \(k\), so \(x/y\) has the same value whatever \(x\) and \(y\) are, provided they satisfy the proportionality. (\(k = 1\) would force \(y = 0\), which the question excludes.) It does not depend on \(xy\), on \(x\) alone or on \(y\) alone.