About Simplification
Simplification is not arithmetic. It is arithmetic done in the right order. Almost every wrong answer in this topic comes from working strictly from left to right and ignoring the rule that multiplication and division are done before addition and subtraction, with brackets done before anything else.
What you need to understand
- The order of operations is brackets first, then powers and roots, then division and multiplication, then addition and subtraction.
- Multiplication and division have equal priority and are worked from left to right. The same is true of addition and subtraction.
- A bracket is a single quantity, so everything inside it is worked out before it is multiplied or divided.
- A number written immediately before a bracket multiplies the whole bracket, not just the first term inside it.
- Order changes the answer: a + b x c is not the same as (a + b) x c, and the two differ by a whole multiple of a.
Formulas to remember
How to work through these questions
- Underline the multiplications and divisions first, work those out, and leave the additions and subtractions until later.
- Work out every bracket before touching anything outside it.
- Rewrite the expression after each step so the work still to be done is always visible.
- Estimate the rough size of the answer first. An expression of twenty plus a product of two small numbers cannot come to several thousand.
- Watch the minus signs when a bracket is removed, because dropping one changes the answer completely.
Mistakes that cost marks
- Working strictly from left to right, which is the commonest single cause of a wrong answer here.
- Multiplying only the first term inside a bracket by the number outside it.
- Subtracting before multiplying, so that b x (c - d) is treated as b x c - d.
- Adding together the terms that should have been multiplied.
- Losing a minus sign when expanding a bracket.
Worked example
Simplify 12 + 8 x 9 - 6.
- Multiplication comes before addition and subtraction, so work out 8 x 9 first.
- 8 x 9 = 72.
- The expression has now become 12 + 72 - 6.
- Working from left to right through the remaining additions and subtractions, 12 + 72 = 84 and 84 - 6 = 78.
Answer: 78
Practice questions with answers
A few Simplification questions with the full solution shown, so you can see
how the method is applied before you attempt the timed set.
Question 1
Find the value of $12 + 7 \times 10 - 7$.
Answer: Option A — with explanation
Multiplication is done before addition and subtraction.
$7 \times 10 = 70$
$12 + 70 - 7 = 75$.
Common mistakes
- worked from left to right and multiplied after adding: 183
- multiplied by the difference instead of subtracting at the end: 33
- added the two numbers that should have been multiplied: 22
Question 2
Simplify: $11 \times 3 + 6 \times 3 - 6$
Answer: Option C — with explanation
Both products are worked out first, then the additions and subtractions.
$11 \times 3 = 33$ and $6 \times 3 = 18$
$33 + 18 - 6 = 45$.
Common mistakes
- added the last number instead of subtracting it: 57
- multiplied the sums instead of the separate numbers: 120
- added all the numbers and ignored the multiplications: 17
Question 3
Simplify: $(4 + 7) \times 7 - 4$
Answer: Option B — with explanation
The bracket is worked out first because it is a single quantity.
$(4 + 7) = 11$
$11 \times 7 - 4 = 73$.
Common mistakes
- added the multiplier instead of multiplying by it: 14
- subtracted before multiplying: 33
- opened the bracket without multiplying the whole sum: 49
Question 4
Work out: $15 + 10 \times 7 - 1$
Answer: Option C — with explanation
Multiplication is done before addition and subtraction.
$10 \times 7 = 70$
$15 + 70 - 1 = 84$.
Common mistakes
- worked from left to right and multiplied after adding: 174
- multiplied by the difference instead of subtracting at the end: 75
- added the two numbers that should have been multiplied: 31
Question 5
Work out: $3 \times 11 + 10 \times 11 - 16$
Answer: Option A — with explanation
Both products are worked out first, then the additions and subtractions.
$3 \times 11 = 33$ and $10 \times 11 = 110$
$33 + 110 - 16 = 127$.
Common mistakes
- added all the numbers and ignored the multiplications: 19
- added the last number instead of subtracting it: 159
- multiplied the sums instead of the separate numbers: 278
Frequently asked questions
Does multiplication always come before addition?
Yes, unless brackets say otherwise. Multiplication and division are always completed before addition and subtraction.
Are multiplication and division done in any order?
No. They have equal priority, so they are done from left to right. In 24 divided by 4 multiplied by 2, the division happens first and the answer is 12.
What happens with a number just before a bracket?
It multiplies everything inside the bracket. So 3 x (4 + 5) is 3 x 9 = 27, and not 3 x 4 + 5.
How can I avoid careless mistakes?
Work out and cross out the multiplications first, rewrite what is left, and only then do the additions and subtractions. Most errors in this topic come from skipping that first step.
