Hard GCSE Priority of operations Questions

Look at the answer given

The student claims the answer is $5$. Let us see where $5$ could come from.

Where $5$ comes from

$5$ is what you get if you add first: $(4+6)\div 2$.

Check that

$10 \div 2 = 5$, so the student added before dividing.

But there are no brackets

The original has no brackets, so we cannot add first.

Recall the rule

BIDMAS says division comes before addition.

Do the division first

So work out $6\div 2$ first.

Evaluate it

$6 \div 2 = 3$.

Now add

Add the $4$ to the result.

Evaluate

$4+3 = 7$.

So the correct answer is

The right answer is $7$, not $5$.

State the mistake

The student's error was doing the addition before the division.

What would justify their working

Only brackets around $4+6$ would make adding first correct.

But those brackets are not there

Without brackets, division must come first.

Confirm the correct value

Dividing first then adding gives $7$.

State the answer

So the mistake was adding before dividing, and the correct answer is $7$.

List the required features

The expression must contain a pair of brackets, a power and a square root, and equal $10$.

Start with a bracket and a power

A bracket with a square is an easy building block.

Work out the bracket

$1+4 = 5$.

Apply the power

$5^{2} = 25$.

Decide how to reach $10$

We need to subtract something to bring $25$ down to $10$.

Find the amount

We must subtract $15$.

Make $15$ a square root

$15² = 225$, so $\sqrt{225} = 15$. This adds the required root.

Write the expression

This has a bracket, a power and a root.

Evaluate the bracket-power

The first part is $25$.

Evaluate the root

The root part is $15$.

Subtract

$25 - 15 = 10$, exactly as required.

Check the features are present

Bracket, power and root all appear.

Confirm the value

The expression equals $10$.

Note other answers exist

This is one possible expression; others also work.

State the expression

So a valid expression is $(1+4)² - \sqrt{225} = 10$.

Find the reciprocal

The reciprocal of $2/5$ is found by turning it upside down.

Write the product

Multiply the number by its reciprocal.

Multiply the numerators

The tops give $2 \times 5 = 10$.

Multiply the denominators

The bottoms give $5 \times 2 = 10$.

Write the fraction

So the product is $10/10$.

Simplify

$10/10 = 1$.

So the product is

A number times its reciprocal is $1$ here.

Now the general case

Take any fraction $a/b$ and its reciprocal $b/a$.

Multiply the tops

The numerator becomes ab.

Multiply the bottoms

The denominator becomes ba.

They are the same

Multiplication can be done in any order, so top and bottom match.

So the fraction is

A number over itself is $1$.

Condition

This works for any non-zero number.

Why zero is excluded

Zero has no reciprocal because you cannot divide by $0$.

State the conclusion

So the product of any non-zero number and its reciprocal is always $1$.

Innermost bracket first

Work from the inside out.

Rewrite the calculation

Put the $4$ into the square brackets.

First power inside

$4^{2} = 16$.

Second power inside

$2^{3} = 8$.

Subtract inside

$16 - 8 = 8$, so the square bracket is $8$.

Rewrite

Now divide $96$ by the bracket value.

Divide

$96 \div 8 = 12$.

Note the order

Bracket, powers inside, subtract, then divide by the whole bracket.

Important point

We divide by the whole bracket ($8$), not just one part.

Check

$8 \times 12 = 96$, confirming the division.

Confirm the powers

The two powers were $16$ and $8$.

Confirm the bracket

The square bracket was $8$.

Restate

The final division gives $12$.

Sense check

The steps combine to $12$.

State the answer

So the answer is $12$.

Powers first

Do all the indices before multiplying, dividing, adding or subtracting.

Rewrite the calculation

Put the power values in.

Identify the multiply and divide

These come before adding and subtracting.

The multiplication

$2 \times 4 = 8$.

The division

$8 \div 2 = 4$.

Rewrite

Now only $+$ and $-$ remain.

Work left to right

First $2+8 = 10$.

Then subtract

$10 - 4 = 6$.

Recall the order

Indices, then multiply/divide, then add/subtract.

Check the first power

$2^{2} = 4$.

Check the second power

$2^{3} = 8$.

Check the multiply

The multiplication was $8$.

Check the divide

The division was $4$.

Confirm the total

The running total is $6$.

State the answer

So the answer is $6$.