← Maths

Word Problems Based on Numbers

Competitive Mathematics · Number System

7 Part1 worked examples5 practice questions

Word problems need more reading than maths. “Three-fourths of a number is 24 less than two-thirds of it” — turn it into one equation and half the job is done. We first learn to translate the words, then the four kinds of problems: operations on numbers, numbers made from digits, heads-and-legs problems, and marking-scheme / notes problems.

1

Translating the words

A symbol for every phrase

The common phrases and their mathematical form.

In the problemIn maths
a number
is / will be / equals
of
more than
less than (or subtracted from )
difference between and bigger − smaller
exceeds by
times
double, triple
a number and its reciprocal
ratio of to

Three steps: (1) give the unknown a letter (“let the number be ”); (2) turn each phrase into a symbol using the table and build one equation; (3) solve and then give what the question asked (the number, half of it, the larger one…), not just .

Example 1 · Basic
Five times a number equals the number plus 36. Find the number.
Solution, step by step
  1. 1Let the number be : .
  2. 2 ⟹
Answer 9
BasicTry it yourself first
7 added to 3 times a number gives the same as 25 added to 1 times it. What is the number?
StandardTry it yourself first
4 added to 5 times a number gives the same as 46 added to 2 times it. What is the number?
StandardTry it yourself first
12 added to 7 times a number gives the same as 48 added to 3 times it. What is the number?
BasicTry it yourself first
A number divided by 13 gives quotient 8 and remainder 5. What is the number?
StandardTry it yourself first
A number divided by 27 gives quotient 36 and remainder 14. What is the number?
2

Operations on numbers

Fractions · difference · sum-product · consecutive

Fractional parts, “more than”, reciprocals from sum and product, and consecutive numbers.

Fractional parts and “more than”

One equation, one unknown

Questions on a “part of a number” turn straight into linear equations.

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Sum, difference and product

Find both numbers or use an identity

Questions on the sum, difference and product of two numbers, and a number plus its reciprocal.

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Consecutive numbers

Take the middle one as

How to write consecutive natural, even and odd numbers with a letter.

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3

Numbers made from digits

Tens, units and reversing

Writing a two-digit number as , what reversing does, and conditions on digits.

The algebraic form

and reversed

Writing two- and three-digit numbers with letters.

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Finding a number from conditions on digits

Two conditions, two letters

Form two equations: one from the sum or difference of the digits, the other from the number and its reverse.

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4

Heads-and-legs, marking schemes and notes

Assume all are one kind

When the total count and the total value (legs, marks, rupees) of two kinds of things are given.

Heads and legs

Two kinds of creatures or vehicles

Two equations, or the “all the same kind” trick.

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Marking schemes and notes

Wrong answers / items of different value

Marks gained for right and lost for wrong — or notes of two values: the same “all one kind” trick.

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Subtracting a number from a ratio

The same number from both terms

When subtracting (or adding) the same number to both terms makes a new ratio.

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5

Traps and fast tricks

Slips · fast methods

How marks leak in word problems and how to save time.

The commonest mistakes

Six traps

The slips that cost marks.

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Fast tricks

Test the options · formulas

Solving by testing the options directly, and formulas worth remembering.

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6

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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7

Revision — formula cards

Flip each card and test yourself

Read the front, think, then flip.

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