The sum of two numbers is 40 and their difference is 8. What is the larger number?

24. The larger number is 40 + 82 = 24 (and the smaller is 40 - 82 = 16).

A number is as much greater than 31 as it is less than 75. Find the number.

53. The number is the midpoint of 31 and 75: 31 + 752 = 53.

The sum of three consecutive even numbers is 234. What is the smallest of them?

76. Three consecutive even numbers average to the middle one: 234 3 = 78. So the numbers are 76, 78 and 80, and the smallest is 76.

A two-digit number equals 4 times the sum of its digits. When 18 is added to the number, its digits are reversed. Find the number.

24. Let the number be 10t + u. '4 times the digit sum' gives 10t + u = 4(t + u), i.e. u = 2t. Adding 18 reverses the digits: 9t + 18 = 9u, so u - t = 2. Then 2t - t = 2 gives t = 2, u = 4 and the number 24.

If the numerator of a fraction is increased by 20% and the denominator decreased by 10%, the fraction becomes 1621. What was the original fraction?

47. If the fraction is ab, then 1.2a0.9b = ab 43 = 1621. So ab = 1621 34 = 47.

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Quantitative Aptitude · Numbers & Simplification

Problems on Numbers — Questions & Answers

These are word problems that turn a description of one or more unknown numbers into equations: sums and differences, digit reversal, consecutive integers, and ratios. The skill is translating each sentence faithfully.

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Key formulas

A two-digit number with tens digit $t$ and units digit $u$ is $10t + u$ ; reversing gives $10u + t$
Difference between a two-digit number and its reverse is $9(t-u)$
Three consecutive integers: $n-1,\; n,\; n+1$ — their sum is $3n$

5 solved Problems on Numbers questions

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01
The sum of two numbers is 40 and their difference is 8. What is the larger number?
4 steps · reveal one at a time
1. Step 1: Understand the problem
  We are given two pieces of information about two numbers: their sum is $40$ and their difference is $8$ . Our goal is to find the value of the larger of these two numbers.
2. Step 2: Apply the formula
  To find the larger number when you know both the sum and the difference of two numbers, you can use the formula: $\text{Larger Number} = (\text{Sum} + \text{Difference}) / 2$ .
3. Step 3: Calculate the larger number
  Substitute the given values into the formula: $\text{Larger Number} = (40 + 8) / 2$ . First, calculate the sum: $40 + 8 = 48$ . Then, divide by $2$ : $48 / 2 = 24$ .
4. Step 4: The answer
  So the answer is (A) 24. The larger number is $24$ , because $(24 + 16 = 40)$ and $(24 - 16 = 8)$ , which matches the given conditions.
02
A number is as much greater than 31 as it is less than 75. Find the number.
5 steps · reveal one at a time
1. Step 1: Understand the relationship
  The problem states that a number is 'as much greater than 31 as it is less than 75'. This means the number is exactly halfway between 31 and 75.
2. Step 2: Identify the method
  To find a number that is exactly halfway between two other numbers, we need to calculate their average or midpoint. This is done by adding the two numbers and dividing by 2.
3. Step 3: Calculate the midpoint
  We need to find the average of 31 and 75. The formula for the midpoint is $\frac{\text{Number 1} + \text{Number 2}}{2}$ . So, we calculate $\frac{31 + 75}{2}$ .
4. Step 4: Perform the calculation
  First, add 31 and 75: $31 + 75 = 106$ . Then, divide the sum by 2: $106 \div 2 = 53$ . The number is 53.
5. Step 5: The answer
  So the answer is (C) 53. This number is $53 - 31 = 22$ greater than 31, and $75 - 53 = 22$ less than 75, confirming it's the correct midpoint.
03
The sum of three consecutive even numbers is 234. What is the smallest of them?
4 steps · reveal one at a time
1. Step 1: Find the Middle Number
  For any set of consecutive numbers, including consecutive even numbers, their average is always the middle number. To find the average, divide the sum by the count of numbers. Here, we have 3 numbers.
2. Step 2: Calculate the Middle Number
  The sum of the three consecutive even numbers is 234. Since there are 3 numbers, the middle number is $234 \div 3 = 78$ . This means 78 is the second of the three numbers.
3. Step 3: Identify the Smallest Number
  Since 78 is the middle even number, the even number immediately before it is $78 - 2 = 76$ . The even number immediately after it is $78 + 2 = 80$ . So the three numbers are 76, 78, and 80.
4. Step 4: The answer
  So the answer is (A) 76. From the set of numbers 76, 78, and 80, the smallest number is 76.
04
A two-digit number equals 4 times the sum of its digits. When 18 is added to the number, its digits are reversed. Find the number.
5 steps · reveal one at a time
1. Step 1: Set up the number
  Let's represent the two-digit number using its tens digit $t$ and units digit $u$ . So, the number can be written as $10t + u$ . Remember, $t$ and $u$ are single digits.
2. Step 2: Apply the first rule
  The question states the number equals 4 times the sum of its digits. We can write this as an equation: $10t + u = 4(t + u)$ . Simplifying this gives us $10t + u = 4t + 4u$ , which further simplifies to $6t = 3u$ , or $u = 2t$ .
3. Step 3: Apply the second rule
  Next, we know that when 18 is added to the number, its digits are reversed. The reversed number is $10u + t$ . So, $(10t + u) + 18 = 10u + t$ . Rearranging this equation gives $18 = 9u - 9t$ , which simplifies to $2 = u - t$ .
4. Step 4: Solve for the digits
  Now we have two simple equations: $u = 2t$ and $u - t = 2$ . Substitute the first equation into the second: $(2t) - t = 2$ . This gives $t = 2$ . Then, substitute $t=2$ back into $u=2t$ to find $u=4$ . The digits are $t=2$ and $u=4$ .
5. Step 5: The answer
  So the answer is (C) 24. The number is $10t + u = 10(2) + 4 = 24$ . This number satisfies both conditions: $24 = 4(2+4)$ and $24+18=42$ (digits reversed).
05
If the numerator of a fraction is increased by 20% and the denominator decreased by 10%, the fraction becomes $\dfrac{16}{21}$ . What was the original fraction?
5 steps · reveal one at a time
1. Step 1: Set up the equation
  Let the original fraction be $\dfrac{a}{b}$ . Increasing the numerator by 20% makes it $1.2a$ . Decreasing the denominator by 10% makes it $0.9b$ . The new fraction is $\dfrac{1.2a}{0.9b}$ , which equals $\dfrac{16}{21}$ .
2. Step 2: Simplify the expression
  We can rewrite the new fraction as $\dfrac{1.2}{0.9} \times \dfrac{a}{b}$ . Simplifying the ratio $\dfrac{1.2}{0.9}$ gives us $\dfrac{12}{9}$ , which reduces to $\dfrac{4}{3}$ . So, we have $\dfrac{4}{3} \times \dfrac{a}{b} = \dfrac{16}{21}$ .
3. Step 3: Solve for original fraction
  To find the original fraction $\dfrac{a}{b}$ , we need to isolate it. Divide both sides of the equation by $\dfrac{4}{3}$ . This means multiplying by its reciprocal, $\dfrac{3}{4}$ . So, $\dfrac{a}{b} = \dfrac{16}{21} \times \dfrac{3}{4}$ .
4. Step 4: Calculate the result
  Now, perform the multiplication: $\dfrac{16}{21} \times \dfrac{3}{4}$ . We can simplify by canceling common factors. $16$ divided by $4$ is $4$ , and $21$ divided by $3$ is $7$ . This leaves us with $\dfrac{4}{7}$ .
5. Step 5: The answer
  So the answer is (C) $\dfrac{4}{7}$ . This matches our calculated original fraction after reversing the percentage changes.

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