If a number is decreased by 14 and then divided by 2, the result is 6. What would be the result if 2 is subtracted from the number and it is then divided by 8?

3. Let the number be x. The first condition says (x - 14) / 2 = 6. So x - 14 = 2 × 6 = 12. That gives x = 12 + 14 = 26. Now apply the second operation: subtract 2 from x and divide by 8. (26 - 2) / 8 = 24 / 8 = 3. Check: (26 - 14) / 2 = 12 / 2 = 6, which matches the first condition.

A number, when multiplied by 13, is increased by 372. What is the number?

31. Let the number be x. Multiplying it by 13 increases it by 372. So 13x = x + 372. That simplifies to (13 - 1)x = 372. Solving gives x = 372 / 12 = 31. Check: 13 × 31 = 403, and 403 - 31 = 372, which matches.

The sum of twice a number and three times 31 is 233. What is the sum of thrice the number and two times 31?

272. Let the number be x. Twice the number plus three times 31 equals 233. So 2x + 3×31 = 233. That gives 2x = 233 - 93 = 140. Solving gives x = 140 / 2 = 70. Now find thrice the number plus two times 31: 3×70 + 2×31 = 210 + 62 = 272. Check: 2×70 + 3×31 = 140 + 93 = 233, which matches the given sum.

If one-half of one-third of a number is 32, what is three-fourths of that number?

144. Let the number be x. 1/2 of 1/3 of x equals 32. So 1/2 × 1/3 × x = 32, which is 1/6x = 32. Solving gives x = 32 / (1/6) = 192. Now find 3/4 of x: 3/4 × 192 = 144. Check: 1/2 × 1/3 × 192 = 1/6 × 192 = 32, which matches.

A two-digit number exceeds the number formed by reversing its digits by 36. If the sum of its digits is 6, what is the original number?

51. Let the tens digit be a and the units digit be b. The original number is 10a + b, and the reversed number is 10b + a. Fact 1: the digits sum to 6, so a + b = 6. Fact 2: the number exceeds its reverse by 36, so (10a+b) - (10b+a) = 36. That simplifies to 9(a - b) = 36, so a - b = 4. Adding a + b = 6 and a - b = 4 gives 2a = 10, so a = 5. Then b = 6 - 5 = 1. So the original number is 10×5 + 1 = 51. Check: reversing gives 10×1 + 5 = 15, and 51 - 15 = 36, which matches.

In a two-digit number, the unit's and ten's digits are 6 and 4 respectively. If the digits are interchanged, the new number is how much more than the original?

18. Let the tens digit be a = 4 and the units digit be b = 6. The original number is 10a + b = 46. Interchanging the digits gives the new number 10b + a = 64. The increase is (10b + a) - (10a + b) = 9(b - a). Here b - a = 2, so the increase is 9 × 2 = 18. Check: 46 + 18 = 64, and the reversed number is 64, which matches.

If a and b are two numbers such that their sum is S and their difference is D, the larger number is expressed as what?

(S+D)/2. Let the larger number be a and the smaller be b. Fact 1: a + b = S, their sum. Fact 2: a - b = D, their difference. Adding these two equations: (a+b) + (a-b) = S + D. The b terms cancel, leaving 2a = S + D. Dividing both sides by 2 gives a = (S+D)/2. Check: then b = S - a = (S-D)/2, and a - b = D, confirming a is the larger number.

Consider this data-sufficiency question: what is the two-digit number? Statement I: the sum of its digits is 9. Statement II: the difference between its digits is 1. Which of these is correct?

The data in both statements together is genuinely NOT sufficient to determine a unique number. Let the two digits be p and q, with p the larger one. Statement I gives p + q = 9. Statement II gives p - q = 1. Adding these equations: 2p = 10, so p = 5. Subtracting them: 2q = 8, so q = 4. So the digits are 5 and 4, but we don't know which is tens and which is units. If 5 is the tens digit, the number is 54. If 4 is the tens digit, the number is 45. Both numbers fit both statements, so together they still don't give a unique number. Check: 54 and 45 both have digits 5 and 4, summing to 9 and differing by 1.

Problems on Numbers: Formulas, Shortcuts and Solved Examples

Number problems turn a sentence into an equation.

This lesson gives each rule, the reason it works and seven worked examples. It also lists the traps that cost marks.

The idea in brief

Unknown: call it x and translate each phrase.
Two-digit number: 10a + b.
Digit swap: changes value by 9(a − b).
Sum and difference: larger = (S + D)/2.
Chain: undo it from the end.

What Number Problems Ask

What are these questions about?

They describe a hidden number in words. You turn the words into an equation and solve.

Some ask for two-digit numbers and their digits.

What is the key move?

Call the hidden number x. Then write each phrase as maths.

“Decreased by 14” means x − 14. “Divided by 2” means ÷ 2. “Twice” means 2x.

Write the chain step by step. Then undo it from the end to find x.

How are two-digit numbers written?

A two-digit number has a tens digit a and a units digit b.

Its value is 10a + b, not a + b.

Check 1

The value of a two-digit number with digits a and b is a + b.

Core Rules and Why They Work

How do I find two numbers from their sum and difference?

Larger = (S + D)/2. Smaller = (S − D)/2.

Why it works: adding the two equations a + b = S and a − b = D gives 2a = S + D. Subtracting them gives 2b = S − D.

What is the value of a two-digit number?

Number = 10a + b. Reversed number = 10b + a.

Why it works: place value makes the tens digit count ten times.

What happens when the digits swap?

The difference is 9 × (the difference between the digits).

Why it works: (10a + b) − (10b + a) = 9a − 9b = 9(a − b).

Swapping digits changes the value by nine times the digit gap.

How do I handle “a fraction of a fraction”?

Multiply the fractions, then solve.

One-half of one-third of x is x/6.

Why it works: “of” means multiply. So the chain collapses into one fraction of x.

What if a number times k is more than the number?

kx = x + extra, so x = extra ÷ (k − 1).

Why it works: kx − x = (k − 1)x equals the extra amount.

Check 2

Swapping the digits of 63 changes its value by a multiple of 9.

Seven Solved Examples

Easy, step by step: a number is decreased by 14 and then divided by 2 to give 6. What is (number − 2) ÷ 8?

Step 1: (x − 14)/2 = 6, so x − 14 = 12.

Step 2: x = 26.

Step 3: (26 − 2)/8 = 24/8.

The answer is 3.

Easy, step by step: a number times 13 is 372 more than the number. Find it.

Step 1: 13x − x = 372.

Step 2: 12x = 372.

Step 3: x = 372 ÷ 12 = 31.

The number is 31.

Medium, step by step: twice a number plus three times 31 is 233. Find thrice the number plus two times 31.

Step 1: 2x + 93 = 233, so 2x = 140.

Step 2: x = 70.

Step 3: 3 × 70 + 2 × 31 = 210 + 62.

The answer is 272.

Medium, step by step: one-half of one-third of a number is 32. Find three-fourths of it.

Step 1: one-half of one-third is 1/6, so x/6 = 32.

Step 2: x = 192.

Step 3: three-fourths of 192 = 144.

The answer is 144.

Medium, step by step: a two-digit number is 36 more than its reverse. Its digits sum to 6. Find it.

Step 1: the difference is 9(a − b) = 36, so a − b = 4.

Step 2: a + b = 6, so a = 5 and b = 1.

Step 3: the number is 51.

Check: 51 − 15 = 36. The number is 51.

Exam level, step by step: the units digit is 6 and the tens digit is 4. By how much does the number grow if the digits swap?

Step 1: the original number is 46.

Step 2: the swapped number is 64.

Step 3: 64 − 46 = 18.

Shortcut: 9 × (6 − 4) = 18. The answer is 18.

Exam level, step by step: the sum of two numbers is S and their difference is D. Name the larger one.

Step 1: write a + b = S and a − b = D.

Step 2: add them to get 2a = S + D.

Step 3: divide by 2.

The larger number is (S + D)/2.

Check 3

In example 5, the digit difference is 4 because 36 divided by 9 is 4.

Common Traps and Shortcuts

Which traps cost marks?

Writing a two-digit number as a + b.
Reading “decreased by 14 then divided by 2” as x − 14/2.
Using the first number in the second chain.
Forgetting to answer what was asked, not just x.
Mixing up the larger and smaller number.

Which shortcuts save time?

Undo the chain from the end.
Use the digit gap rule 9(a − b).
Fold “fraction of fraction” into one fraction.
Test each option in the equation.

When is a data-sufficiency question not solvable?

Two statements may still leave several answers. A digit sum of 9 and a digit gap of 1 give 54 or 45.

So both statements together are not enough.

Check 4

A digit sum of 9 and a digit gap of 1 fix one unique number.

0/0 understanding checks correct

Was this helpful?

Take a Quantitative Aptitude QuizRandom practice question from this subject →

Practice 89 MCQs on this topic

Set 1 Set 2 Set 3 Set 4