About Problems on Numbers
These are word problems that reduce to a single linear equation. The method never varies: name the unknown, turn the sentence into an equation, and then work backwards through the operations the question describes.
What you need to understand
- The sum and the difference of two numbers together determine both of them.
- Adding the sum and the difference cancels the smaller number, and subtracting the difference cancels the larger one.
- Working backwards means undoing the operations in reverse order, so the addition is undone before the multiplication.
- A two digit number is ten times its tens digit plus its units digit.
- Interchanging the digits of a two digit number changes it by nine times the gap between the digits.
- Whatever the question, the answer can be checked by putting it back into the sentence.
Formulas to remember
How to work through these questions
- Read the sentence and write it as an equation, using a single letter for the unknown.
- Move backwards through the operations, undoing each one in the reverse order.
- For digit questions, write the number as 10a + b and turn every statement into a statement about a and b.
- Use the sum and difference formulas when both are given, rather than writing two equations.
- Substitute the answer back into the original sentence as a check.
Mistakes that cost marks
- Dividing before subtracting when working backwards, which reverses the order of the operations.
- Giving the smaller number when the question asks for the larger.
- Forgetting that the tens digit is worth ten times its face value.
- Treating the sum as though it were one of the numbers.
- Confusing the gap between the digits with nine times that gap.
Worked example
The sum of two numbers is 121 and their difference is 47. Find the larger number.
- Adding the sum and the difference removes the smaller number entirely.
- Larger = (121 + 47) / 2 = 168 / 2 = 84.
- The smaller number is (121 - 47) / 2 = 74 / 2 = 37.
- Check: 84 + 37 = 121 and 84 - 37 = 47, so both conditions hold.
Answer: 84
Practice questions with answers
A few Problems on Numbers questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
The sum of two numbers is 77, and the larger exceeds the smaller by 29. Find the larger number.
Answer: Option C — with explanation
Adding the sum and the difference removes the smaller number.
Larger = (sum + difference) / 2 = (77 + 29) / 2 = 53.
The smaller number is (77 - 29) / 2 = 24.
Common mistakes
- added the difference instead of halving the sum: 106
- gave the smaller number: 24
- gave the sum instead of one of the numbers: 77
Question 2
A number multiplied by 11 and then increased by 7 gives 84. Find the number.
Answer: Option D — with explanation
Undo the operations in reverse order. First take away 7: 84 - 7 = 77.
Then divide by 11: 77 / 11 = 7.
Common mistakes
- added one instead of subtracting the constant: 8
- added the constant a second time: 14.64
- divided before subtracting: 7.64
Question 3
The sum of the digits of a two digit number is 11. If the digits are interchanged, the new number is 9 more than the original. Find the original number.
Answer: Option D — with explanation
Let the number be 10a + b with a + b = 11 and b - a = 9 / 9 = 1.
Solving gives b = 6 and a = 5, so the number is 56.
Check: the digits add to 11, and the reversed number 65 exceeds 56 by 9.
Common mistakes
- applied the difference only once instead of nine times the digit gap: 47
- gave the number formed by interchanging the digits: 65
- used the digit sum in the tens place: 116
Question 4
The sum of two numbers is 94 and their difference is 20. Find the larger number.
Answer: Option B — with explanation
Adding the sum and the difference removes the smaller number.
Larger = (sum + difference) / 2 = (94 + 20) / 2 = 57.
The smaller number is (94 - 20) / 2 = 37.
Common mistakes
- gave the sum instead of one of the numbers: 94
- gave the smaller number: 37
- added the difference instead of halving the sum: 114
Question 5
36 added to 7 times a number gives 134. Find the number.
Answer: Option B — with explanation
Undo the operations in reverse order. First take away 36: 134 - 36 = 98.
Then divide by 7: 98 / 7 = 14.
Common mistakes
- forgot to divide by the multiplier: 98
- added the constant to the number itself: 50
- added the constant a second time: 55.14
Frequently asked questions
Why do we add the sum and the difference?
Because the smaller number carries a plus sign in the sum and a minus sign in the difference, so adding the two statements removes it and leaves twice the larger number.
How do I find a number that was multiplied and then increased?
Work backwards: subtract the increase first and then divide by the multiplier. Reversing the order of those two steps gives a wrong answer.
What is the value of a two digit number?
Ten times the tens digit plus the units digit. The number 47 is 10 x 4 + 7, because the 4 stands for four tens.
What happens when the digits of a two digit number are interchanged?
The number changes by nine times the gap between the digits. For 47 the digits differ by 3, and the reversed number 74 is 27 more, which is 9 x 3.
