Problems on Numbers Questions and Answers

Answer: B

The number is 280.

Let the number be \(N\). The mistaken answer exceeds the correct one by 150:

\[ \begin{aligned} \tfrac{3}{4}N - \tfrac{3}{14}N &= 150 \\ \tfrac{21 - 6}{28}N &= 150 \\ \tfrac{15}{28}N &= 150 \\ N &= 280 \end{aligned} \]

Check: \(\frac{3}{4}\) of 280 is 210 and \(\frac{3}{14}\) of 280 is 60, and \(210 - 60 = 150\). With 500, 180 or 160 the difference would not be 150; for example \(\frac{15}{28}\) of 500 is about 268.

Answer: D

The sum is 2604.

The second number is 12% of 100, which is 12. Its cube is \(12^{3} = 1728\), and that equals two thirds of the first number:

\[ \begin{aligned} \tfrac{2}{3}x &= 1728 \\ x &= 1728 \times \tfrac{3}{2} = 2592 \\ x + 12 &= 2604 \end{aligned} \]

2640 and 2408 come from slips in the multiplication by 3/2, and 2426 from adding the wrong second number. Only 2604 satisfies both conditions.

Answer: B

The number is 105.

Let the number be \(x\). Two thirds of it exceeds three fifths of it by 7:

\[ \begin{aligned} \frac{2x}{3} - \frac{3x}{5} &= 7 \\ \frac{10x - 9x}{15} &= 7 \\ x &= 105 \end{aligned} \]

Check: \(\frac{2}{3} \times 105 = 70\) and \(\frac{3}{5} \times 105 = 63\), which differ by 7. The other options give differences of 4, 10 and 8.

A quicker route: the difference of the two fractions is \(\frac{2}{3} - \frac{3}{5} = \frac{1}{15}\) of the number, and that equals 7, so the number is \(15 \times 7 = 105\).

Answer: B

The number is 8.

A number added to itself 13 times is the number plus 13 more copies of it, 14 copies in all:

\[ \begin{aligned} n + 13n &= 112 \\ 14n &= 112,\ n = 8 \end{aligned} \]

Check: \(14 \times 8 = 112\). The trap is to use 13 copies, but 112 is not a multiple of 13. 7, 9 and 11 give 98, 126 and 154.

Read literally, adding the number to itself 13 times makes 13 additions to the original, which is why the total is 14 times the number.

Answer: C

The original number is 39.

Let the tens digit be \(a\) and the units digit \(b\). Reversing the digits increases the number, so \(b > a\).

\[ \begin{aligned} (10b + a) - (10a + b) &= 54 \\ 9(b - a) &= 54 \\ b - a &= 6 \\ b + a &= 12 \end{aligned} \]

So \(b = 9\), \(a = 3\), and the number is 39; reversing it gives 93, which is 54 more. 93 is the reversed number, the trap. 75 and 48 have digits summing to 12 but differ from their reversals by 18 and 36.

Answer: A

Let the smaller be \( y \); the greater is \( 4y + 5 \).

\[ y + 4y + 5 = 80 \quad\Longrightarrow\quad 5y = 75 \]

\[ y = 15, \qquad \text{greater} = 65 \]

Answer: D

\[ x - \frac{3x}{8} = 25 \quad\Longrightarrow\quad \frac{5x}{8} = 25 \]

\[ x = 40 \]

Answer: D

Let the number be \(x\)
Then, \(\dfrac{1}{3}\) of \(\dfrac{1}{4}\) of \(x = 15 \Leftrightarrow x =15 \times 12 = 180\)
So, required number
\(\left( \dfrac{3}{10} \times 180 \right) =54\)

Answer: B

Let the middle digit be \(x.\)
Then, \(2x = 10\) or \(x = 5\). So, the number is either \(253 \text{ or } 352.\)
Since the number increases on reversing the digits, so the hundred's digits is smaller than the unit's digit.
Hence, required number \(= 253\)

Answer: C

Let the second number be \(x\). Then, first number \(= 2x\) and third number \(= \dfrac{2x}{3}\)
\(2x + x + \dfrac{2x}{3} = 264\\ \Rightarrow \dfrac{11x}{3} = 264\\ \Rightarrow x = 72\)