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Approximation

Competitive Mathematics · Number System

6 Part2 worked examples11 practice questions

Bank exams ask “approximately how much?”: . Multiplying exactly is a waste of time — the question itself says an exact value is not expected. gives a close answer, and the options are usually so far apart that this estimate picks the right one for sure.

You need two skills: (1) swapping each number for a good friendly one (so the error stays near 1%), and (2) doing the multiplication or division quickly with easy numbers. The rest of the chapter practises these.

1

Rounding basics

Nearest 10, 100, decimals

The rule for bringing a number to a nearby round number, and checking how good an estimate is.

Rounding to the nearest

Next digit 5 or more → round up

Rounding to tens, hundreds, thousands or a decimal place.

Look at the first digit to the right of the place you are rounding to. If it is 5 or more, raise that place by one; if 4 or less, keep it; then make everything to the right zero. Rounding 4,876 to the nearest hundred: the tens digit 7 is ≥ 5 → 4,900.

NumberNearest 10Nearest 100Nearest 1
349.62350300350
2,583.42,5802,6002,583
47.4950047
998.51,0001,000999
Example 1 · Basic
Round to the nearest hundred and the nearest unit.
Solution, step by step
  1. 1To the hundred: the tens digit 8 ≥ 5 → 6,300.
  2. 2To the unit: the tenths digit 7 ≥ 5 → 6,286.
Answer and
BasicTry it yourself first
Round 3,476 to the nearest 100.
BasicTry it yourself first
Round 5,249 to the nearest 10.
BasicTry it yourself first
Round 784.6 to the nearest 1.
StandardTry it yourself first
Round 12,950 to the nearest 100.
StandardTry it yourself first
Round 4.37 to the nearest 0.1.
StandardTry it yourself first
Round 99,750 to the nearest 1,000.
AdvancedTry it yourself first
Round 8.1 to the nearest 0.01.
AdvancedTry it yourself first
Round 4,49,950 to the nearest 1,000.

How good is the estimate

Error and percent error

The gap between estimate and exact value, and when an estimate can be trusted.

Checking an estimate

How exact you need to be depends on how far apart the options are. If they are 10–20% apart, an estimate with 1–2% error will do; if they are close (like 1,980 and 2,000), be more careful in swapping the numbers, or refine the estimate once.

Example 2 · Standard
The exact value of is 299.97. What is the percent error of the estimate ?
Solution, step by step
  1. 1Error .
  2. 2Percent error — tiny.
Answer 0.01%
StandardTry it yourself first
The exact value of a calculation is 299.97. Which estimate is better: A = 300 or B = 290?
StandardTry it yourself first
The exact value of a calculation is 1,486. Which estimate is better: A = 1,500 or B = 1,450?
AdvancedTry it yourself first
The exact value of a calculation is 5.04. Which estimate is better: A = 5 or B = 5.1?
2

The rules of approximation

Add-subtract · multiply · divide · percent

Four rules for making big numbers easy, and the percent shortcuts.

Rule 1 — take the nearby number

Addition and subtraction

Replace each number by a nearby round one, then add and subtract normally.

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Rules 2 and 3 — multiplying and dividing

Raise one, lower the other

In a product raise one number and lower the other; in a quotient move both the same way.

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Rule 4 — percent shortcuts

10% · 1% · 25% · 12.5%

Mental shortcuts for the percent of a number.

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3

Fractions, powers, roots and mixed expressions

All the rules together

The fraction-percent table, estimating powers and roots, and long mixed expressions.

Fraction ↔ percent table

Learn it and save time

Percent values of common fractions, on which most estimates rest.

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Estimating powers and roots

Close to a round number

If the number is close to a perfect cube or square, work with that.

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Long mixed expressions

A different rule for each term

When an expression has multiplication, division, percent and powers together.

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4

Traps and fast tricks

Slips · tricks

Where estimates go wrong and how to avoid it.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

The ones that pay off most

A few habits that keep estimates quick and accurate.

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5

The practice ladder

Basic → standard → advanced

New questions of every type, in three levels.

Level 1 — Basic

Direct rules

One-rule questions.

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Level 2 — Standard

Two ideas combined

Exam-level questions.

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Level 3 — Advanced

Many steps and clever thinking

Questions at the hard end of competitive exams.

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6

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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