Fractions and ratio · Number foundations & fractions

Almost every WAEC Paper 2 starts with a fraction simplification “without using tables or a calculator”. It carries easy marks if each step is set out carefully.

Try it

Fractions on barsChange the fractions

2/3 + 1/4 = 11/12with denominator 12: 8/12 + 3/12

first: top = 2first: bottom = 3second: top = 1second: bottom = 4

Pieces must be the same size before you can add them. Cut both bars into 12 pieces (the lowest common multiple of 3 and 4): 2/3 = 8/12 and 1/4 = 3/12. Then 8 + 3 = 11 pieces: 11/12.

To add or subtract, the pieces must be the same size: cut both bars into the lowest common multiple of the denominators. Switch to Divide to see why dividing by a fraction is the same as turning it over and multiplying.

The four operations

Operation	Method
add, subtract	change to a common denominator, then add or subtract the tops
multiply	multiply the tops and multiply the bottoms (cancel first)
divide	turn the second fraction over, then multiply
mixed numbers	change to improper fractions first: $2\frac34 = \frac{11}{4}$

The word “of” means multiply: $\frac13$ of $\frac25$ is $\frac13 \times \frac25 = \frac{2}{15}$ . In BODMAS it counts with the multiplications.

Order of operations

Brackets first, then powers, then division and multiplication, then addition and subtraction (BODMAS). In a big fraction, work out the top and the bottom separately, then divide.

Worked example · WAEC 2017

WAEC 2017 · Paper 2 · Q1 (a)

Simplify: $\dfrac{2\frac12 + 1\frac34 \div 1\frac25}{2\frac14 - 1\frac12}$ .

The top: divide before adding
Division first: $1\frac34 \div 1\frac25 = \frac74 \div \frac75 = \frac74 \times \frac57 = \frac54$ . Then add: $2\frac12 + \frac54 = \frac{10}{4} + \frac54 = \frac{15}{4}$ .
Think first. Which comes first in $2\frac12 + 1\frac34 \div 1\frac25$ ?
The bottom
$2\frac14 - 1\frac12 = \frac94 - \frac64 = \frac34$ .
Divide top by bottom
$\frac{15}{4} \div \frac34 = \frac{15}{4} \times \frac43 = 5$

More: simplifying fractions

“A fraction of the remainder”

Many word problems spend fractions of an income one after another. Keep track of what’s left at each step: a fraction “of the remainder” applies to what’s left, not to the whole.

A fraction of the remainderTake the fraction of what's left, not of the whole

Worked example · WAEC 2020

WAEC 2020 · Paper 2 · Q8 (a)

Ms. Maureen spent $\frac14$ of her monthly income at a shopping mall, $\frac13$ at an open market and $\frac25$ of the remaining amount at a mechanic workshop. If she had ₦225,000.00 left, find: (i) her monthly income; (ii) the amount spent at the open market.

The first two spends
$\frac14 + \frac13 = \frac{7}{12}$ , so $\frac{5}{12}$ is left.
Think first. What fraction of her income is left after the mall and the market?
Two-fifths of the remainder
The workshop takes $\frac25$ of $\frac{5}{12} = \frac16$ of her income. Left: $\frac{5}{12} - \frac16 = \frac{3}{12} = \frac14$ .
(i) The income
$\frac14$ of her income is ₦225,000, so her income is $4 \times 225\,000 =$ ₦900,000.
(ii) The open market
$\frac13 \times 900\,000 =$ ₦300,000.

More: fractions of the remainder

Ratio

To share an amount in the ratio $a : b : c$ , add the parts, find one part, then multiply.

To combine $p : q$ and $q : r$ , make the $q$ parts equal. With $p : q = 2 : 3$ and $q : r = 4 : 5$ : multiply the first by 4 and the second by 3 to get $8 : 12$ and $12 : 15$ , so $p : q : r = 8 : 12 : 15$ . If a ratio is given in fractions, multiply through by the LCM of the denominators to make whole numbers first.

Combining two ratiosMake the shared letter the same size in both

More: ratio

Average speed

A speed is a rate: distance per unit of time, such as kilometres per hour (km/h). For a journey in several parts, the average speed is

\text{average speed} = \frac{\text{total distance}}{\text{total time}}

Speed, distance, timeD = S × T, S = D ÷ T, T = D ÷ S

Average speedTotal distance ÷ total time, not the average of the speeds

It is not the average of the speeds, because the slower part takes longer. For example, drive 120 km at 60 km/h and come back at 40 km/h:

There: ${120 \div 60 = 2}$ hours.
Back: ${120 \div 40 = 3}$ hours.
Total: 240 km in 5 hours.
Average speed: ${240 \div 5 = 48}$ km/h, not ${(60 + 40) \div 2 = 50}$ .

Worked example · WAEC 2010

WAEC 2010 · Paper 2 · Q11 (b)

A motorist travelled $300\text{ km}$ at an average speed of $75\text{ km/h}$ and returned at an average speed of $v\text{ km/h}$ . If his average speed for the whole journey is $60\text{ km/h}$ , find $v$ .

The total distance
- There and back: ${300 + 300 = 600}$ km.
Think first. How far is there and back?
The total time
- ${600 \div 60 = 10}$ hours for the whole journey.
Think first. Use time = distance ÷ speed with the average speed.
The time for each way
- Outward: ${300 \div 75 = 4}$ hours.
- Return: ${10 - 4 = 6}$ hours.
Think first. How long does the outward trip take at 75 km/h?
The return speed
- ${v = 300 \div 6 = 50}$ .
Think first. Speed = distance ÷ time for the return trip.

Factors, primes, HCF and LCM

A prime number has exactly two factors, 1 and itself: 2, 3, 5, 7, 11, 13, …. (1 is not prime.) Every whole number can be written as a product of primes; a factor tree finds them.

A factor treeSplit until every branch ends in a prime

HCF and LCMCommon primes give the HCF; all of them give the LCM

The HCF (highest common factor) multiplies the primes the numbers share.
The LCM (lowest common multiple) multiplies every prime needed, each as often as it appears in any one number.
A perfect square has every prime an even number of times. $72 = 2^3 \times 3^2$ needs one more 2, so the smallest number to multiply 72 by to get a perfect square is 2 (giving $144 = 12^2$ ).

More: factors, primes, HCF and LCM

Fractions and decimals

To compare fractions and decimals, change them all to decimals: $\frac58 = 0.625$ . To divide by a decimal, multiply the top and bottom by a power of 10 until the bottom is whole: $\frac{1}{0.03} = \frac{100}{3}$ . And in a place-value sum, $\frac{3}{1000}$ is the third decimal place: $0.003$ .

More: fractions and decimals

Your turn

WAEC 2023 · Paper 2 · Q7 (a)

(a)
Aman spent $\frac13$ of his income on rent, $\frac16$ of the income in a fast food shop and $\frac34$ of the remaining amount was saved in a bank. If he had ₦125,000.00 left, find: (i) his income; (ii) the amount saved in the bank.
Separate values with commas, e.g. 3, −2

Worked solution (try it first)

(a)

Rent and food take $\frac13 + \frac16 = \frac12$ of his income, leaving $\frac12$ .
He saves $\frac34$ of that: $\frac34 \times \frac12 = \frac38$ .
What's left is $\frac12 - \frac38 = \frac18$ of his income.

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