Approximation — notes, question types & practice · Simplification

01

Overview

When a question starts with 'what approximate value', it is permission to round. The numbers are deliberately awkward, like $399.87$ or $24.98\%$ , and the options sit $5$ to $10$ percent apart. Rounding error stays near one percent, so the nearest option is safe.

02

The rounding rules

Round decimals to the nearest whole number: $49.98$ to $50$ , $15.02$ to $15$ , $23.9$ to $24$ . Snap roots and powers to neighbours: $\sqrt{1224.9}$ sits next to $35^2 = 1225$ , and $(29.98)^2$ is $30^2 = 900$ for all practical purposes.

Rule: Round first, then compute exactly on the rounded numbers. Never carry ugly decimals through five steps.

03

Percentage anchors, learned cold

Fraction	Percent	Fraction	Percent
$1/2$	$50\%$	$1/8$	$12.5\%$
$1/3$	$33.3\%$	$1/9$	$11.1\%$
$1/4$	$25\%$	$1/12$	$8.3\%$
$1/5$	$20\%$	$3/8$	$37.5\%$
$1/6$	$16.7\%$	$2/3$	$66.7\%$

So $16.98\%$ of $600$ reads as one sixth of $600$ , about $100$ . Percentage times round bases is a one-step mental product.

Tip: Convert the percent to its fraction anchor before touching the base number; $37.5\%$ of $480$ is three eighths of $480$ , which is $180$ .

04

Balanced rounding in products

When two rounded numbers multiply, round one up and the other down so the errors cancel. $15.02 \times 23.9$ becomes $15 \times 24 = 360$ , not $15 \times 24$ after rounding both up. One factor down, one factor up, every time.

05

Roots and powers near perfect values

Setters place roots beside perfect powers on purpose. $\sqrt{3968.9}$ is $63$ because $63^2 = 3969$ ; $\sqrt[3]{29791.03}$ is $31$ because $31^3 = 29791$ . Spot the neighbour first, then read the root off it.

06

Size and direction checks

Decide the size region before computing: tens, hundreds or thousands. One decoy option usually lives in each region. For unknowns, direction matters: $18\%$ of $?$ equals $117$ means $? = 117$ divided by $0.18$ , a bigger number; $(?)\%$ of $800$ equals $240$ means $?$ is itself a percentage, here $30$ .

Watch: A question mark before the percent sign asks for a percent; a question mark after 'of' asks for a number. Mixing them flips the answer by a factor of ten.

07

One clean run

$19.98\%$ of $200 + 11.99 \times 6$ : one fifth of $200$ is $40$ , and $12 \times 6 = 72$ , so about $112$ . Round, compute once, choose, move on.

Remember: An approximation question is a reading test, not an arithmetic test. The setter wrote ugly numbers so that rounding is the intended path; the student who multiplies $399.87$ by hand has already lost the time race.

08

Question types you will see

Each type: how to recognise it, the method step by step, and one question to try.

Type 1very common3 practice Q

Mixed chain with percent, multiply and divide

How to spot it:

A question mark with 'approximately equals' and a chain mixing % of, products and divisions of decimals.

\text{round each number, then compute exactly}

Method

Round each decimal to its nearest friendly value.
Replace the percent with its fraction anchor.
Compute the chain with BODMAS order.
Pick the option nearest to the result.

Why it works:

Options sit several percent apart while rounding error stays near one percent.

Try this

$24.98\%$ of $799.8 + 14.98 \times 20.02 \approx ?$

Show solution

$24.98\% \approx 25\% = \dfrac{1}{4}$ , and $799.8 \approx 800$ .
$\dfrac{1}{4} \times 800 = 200$ .
$15 \times 20 = 300$ .
$200 + 300 = 500$ .

Answer

500

Type 2very common3 practice Q

Percentage chains and unknown percent equations

How to spot it:

Two percent terms combined, or an equation where the unknown is itself a percentage.

(?)\% = \dfrac{\text{target} - \text{known part}}{\text{base}} \times 100

Method

Round the base and the targets.
Shift the known numbers to the far side first.
Divide the remainder by the base and scale by 100.

Why it works:

After shifting, the equation is one division on rounded numbers.

Try this

$(?)\%$ of $799.8 + 39.98 \approx 279.9$ . Find the value of $?$ .

Show solution

$799.8 \approx 800$ , $39.98 \approx 40$ , $279.9 \approx 280$ .
$280 - 40 = 240$ is the percent part.
$240 = 30\%$ of $800$ , so $? = 30$ .

Answer

30

Type 3common2 practice Q

Roots and powers inside an approximation

How to spot it:

Near-perfect squares and cubes hide inside the expression, like $(29.98)^2$ or $\sqrt{1224.9}$ .

(a \pm 0.02)^2 \approx a^2,\quad \sqrt{n^2 \pm k} \approx n

Method

Snap each square to the rounded base squared.
Snap each root to the neighbouring perfect power.
Combine the snapped values.

Why it works:

Setters build these expressions within a whisker of perfect powers, so snapping loses almost nothing.

Try this

$(29.98)^2 + \sqrt{1224.9} \approx ?$

Show solution

$29.98 \approx 30$ , so $(29.98)^2 \approx 900$ .
$35^2 = 1225$ , so $\sqrt{1224.9} \approx 35$ .
$900 + 35 = 935$ .

Answer

935

Type 4very common3 practice Q

Find the missing value

How to spot it:

The question mark is a whole term of the equation, not necessarily a percentage.

? = \text{constant side} - \text{evaluated terms}

Method

Evaluate every term that holds no question mark.
Move it across the equals sign with the inverse operation.
Round the result and sanity-check its size.

Why it works:

It is one move of algebra applied to rounded numbers.

Try this

$27.98\%$ of $449.9 + ? \approx 200$ . Find $?$ .

Show solution

$27.98\% \approx 28\%$ and $449.9 \approx 450$ .
$28\%$ of $450 = 126$ .
$? \approx 200 - 126 = 74$ .

Answer

74

Type 5common

Round before dividing

How to spot it:

An awkward division like $399.87 \div 7.99$ , often mixed with other rounded terms.

\text{round divisor and dividend to compatible values}

Method

Round the divisor to a friendly one-digit number.
Round the dividend so the division comes out clean.
Finish the remaining terms and combine.

Why it works:

A clean quotient beats a correct-but-slow long division, because options are far apart.

Try this

$399.87 \div 7.99 + 18.03 \times 4.97 \approx ?$

Show solution

$399.87 \div 7.99 \approx 400 \div 8 = 50$ .
$18.03 \times 4.97 \approx 18 \times 5 = 90$ .
$50 + 90 = 140$ .

Answer

140

09

Formula sheet

Percentage swap

x\% \text{ of } y = y\% \text{ of } x

Near-square root

\sqrt{n^2 + k} \approx n + \frac{k}{2n}

Percent-fraction anchors

12.5\% = \tfrac{1}{8},\quad 16.7\% = \tfrac{1}{6},\quad 33.3\% = \tfrac{1}{3},\quad 37.5\% = \tfrac{3}{8}

10

Shortcuts that save time

⚡ Swap the percentage

x percent of y equals y percent of x. Flip to whichever side has the nicer multiplier.

Example

Find 8% of 125.

Show solution

Swap: $8\%$ of $125 = 125\%$ of $8$ .
$125\%$ of $8 = 8 + \dfrac{1}{4}$ of $8$ .
$8 + 2 = 10$ .

Answer

10

⚡ Balanced rounding for products

Round one factor up and the other down. The two errors cancel and the product stays close.

Example

Approximate 15.02 x 23.9.

Show solution

Round $15.02$ down to $15$ .
Round $23.9$ up to $24$ .
$15 \times 24 = 360$ ; the exact value is $358.98$ .

Answer

360

⚡ Anchors before bases

Turn the percent into its fraction anchor first, then apply it to the rounded base.

Example

Approximate 37.48% of 479.9.

Show solution

$37.5\% = \dfrac{3}{8}$ and $479.9 \approx 480$ .
$\dfrac{3}{8} \times 480 = 3 \times 60$ .
$= 180$ .

Answer

180

11

Mistakes to avoid

Where most students lose marks on this subtopic.

Mistake 01

Rounding every number in a product in the same direction.

Round one factor up and the other down so errors cancel.

Mistake 02

Computing the exact value with long multiplication.

Round, compute on friendly numbers, pick the nearest option; the spacing allows it.

Mistake 03

Treating (?)% as a number and of ? as a percentage.

The percent sign on the question mark makes the unknown a percent; otherwise it is a plain number.

Mistake 04

Forgetting BODMAS order while approximating.

The 'of' product still outranks the division sign in rounded work too.

12

Quick revision

Read this the night before the exam.

Round every number first; compute exactly on rounded values.
Percent to fraction anchor before touching the base.
Products: round one factor up, one down.
Roots sit beside perfect powers; read the neighbour.
Check size region and the direction of the unknown percent.

13

Practice: 12 questions

Sets of 10, mixed across the question types above. Every answer has a step-by-step explanation.

Topic test · 12 questions

Suggested time 8 min · wrong answers go to your mistake notebook automatically.