Mathematics · Lesson 2

Fractions, decimals, approximations, standard form, ratio & percentage error

JAMBWAECNECO

What the syllabus asks

JAMB: Number and numeration: fractions, decimals, approximations and significant figures, standard form, percentage error

WAEC: Number and numeration: fractions (operations and word problems), decimals (conversion, recurring decimals), approximation (decimal places, significant figures, estimation), standard form, limits of accuracy and percentage error

NECO: Number and numeration: fractions (operations and word problems), decimals (conversion, recurring decimals), approximation (decimal places, significant figures, estimation), standard form, limits of accuracy and percentage error

What you will learn

Explain what a fraction is, and use proper fractions, improper fractions and mixed numbers.
Write equivalent fractions, simplify fractions and put fractions in order.
Add, subtract, multiply and divide fractions and mixed numbers, using BODMAS correctly.
Solve word problems involving fractions of quantities.
Convert between fractions, decimals and percentages, including recurring decimals.
Round numbers to given decimal places and significant figures, and estimate calculations.
Write and calculate with numbers in standard form.
Find limits of accuracy for measurements and use them in calculations.
Calculate error and percentage error.

Key points

A fraction is part of a whole cut into equal parts: bottom = how many parts, top = how many you have. Fraction of an amount: ÷ bottom, then × top.
Mixed → improper: (whole × bottom + top) over the bottom. Improper → mixed: divide; the remainder is the new top.
Equivalent fractions: × or ÷ top and bottom by the same number. Simplify by dividing by the HCF.
Compare, add and subtract using a common denominator (the LCM of the bottoms).
Multiply top × top and bottom × bottom (“of” means ×). Divide with Keep, Change, Flip. Mixed numbers become improper first.
BODMAS: Brackets, Of, Division and Multiplication, Addition and Subtraction.
Word problems: ask “a fraction of what?”; remainder = 1 − fraction used; check forwards.
Fraction → decimal: top ÷ bottom; × 100 for a percentage, ÷ 100 to leave one.
Once the fraction is in its simplest form, bottoms with only the primes 2 and 5 terminate. Recurring decimals: let x =, multiply by 10 or 100, subtract.
Rounding: look only at the next digit; 5 or more rounds up. d.p. count after the point; s.f. count from the first non-zero digit.
Estimate by rounding every number to 1 s.f.
Standard form A × 10ⁿ, 1 ≤ A < 10: multiply A’s and add powers; divide A’s and subtract powers; then fix A.
Bounds = value ± half a unit. Percentage error = error ÷ actual value × 100%. For a rounded measurement, maximum percentage error = (half a unit ÷ measured value) × 100%.

Start here: What is a fraction?

Fractions, decimals and percentages appear in nearly every WAEC, NECO and JAMB paper, and you use them every day with naira and kobo. So we start from the very beginning.

Mama cuts a loaf of bread into 4 equal pieces and gives you 1. You have one out of four equal pieces. We write this as 1/4 and say “one quarter”.

Fig. 2.1: A loaf cut into 4 equal pieces. The shaded piece is 1/4 of the loaf.

A fraction shows part of a whole that has been cut into equal parts. It has two numbers:

The bottom number, the denominator, says how many equal parts the whole was cut into: 4.
The top number, the numerator, says how many of those parts you have: 1.

Take 3 of the 4 pieces and you have 3/4 (“three quarters”). The parts must be equal: one big piece and three tiny ones are not quarters.

A fraction of an amount

Worked example 2.1

Your uncle shares ₦1,000 equally among 5 children. What fraction does each child get, and how much money is that?

The whole, ₦1,000, is cut into 5 equal shares, so the bottom is 5.
Each child gets 1 share, so the top is 1: the fraction is 1/5.
To find 1/5 of ₦1,000, divide by the bottom: 1,000 ÷ 5 = 200.

1/5 each, which is ₦200

Worked example 2.2

Find 3/5 of ₦1,000.

One part: divide by the bottom: 1,000 ÷ 5 = 200.
We want 3 parts: multiply by the top: 3 × 200 = 600.

₦600

So, to find a fraction of an amount: divide by the bottom, then multiply by the top.

Try it yourself. A basket holds 24 oranges. How many oranges are 3/8 of the basket?

Show solution

One part: 24 ÷ 8 = 3. Three parts: 3 × 3 = 9 oranges.

Remember: a fraction is part of a whole cut into equal parts. Bottom (denominator) = how many equal parts. Top (numerator) = how many parts you have.

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Practice questions

1. Find 3/4 of ₦2,400.

Show worked answer

Divide by the bottom: 2,400 ÷ 4 = 600. Multiply by the top: 600 × 3 = ₦1,800.

2. (a) Write 4⅝ as an improper fraction. (b) Write 29/6 as a mixed number.

Show worked answer

(a) (4 × 8 + 5)/8 = 37/8. (b) 29 ÷ 6 = 4 remainder 5, so 4⅚.

3. Simplify 42/56 to its simplest form.

Show worked answer

Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. The HCF is 14. 42 ÷ 14 = 3 and 56 ÷ 14 = 4, so 3/4.

4. Simplify 1⅝ + 2¾ − 3⅙.

Show worked answer

Improper: 13/8 + 11/4 − 19/6. LCM of 8, 4 and 6 is 24: 39/24 + 66/24 − 76/24 = 29/24 = 1 5/24.

5. Simplify 2⅔ ÷ 1⅗ × ¾.

Show worked answer

Improper: 8/3 ÷ 8/5 × 3/4. Division and multiplication go left to right. Keep, Change, Flip: 8/3 × 5/8 = 5/3. Then 5/3 × 3/4 = 15/12 = 5/4 = 1¼.

Key words

Fraction: Part of a whole that has been cut into equal parts, e.g. 3/4.
Numerator: The top number: how many parts you have.
Denominator: The bottom number: how many equal parts the whole was cut into.
Proper fraction: Less than 1: top smaller than bottom, e.g. 5/7.
Improper fraction: 1 or more: top equal to or bigger than bottom, e.g. 11/4.
Mixed number: A whole number plus a proper fraction, e.g. 2¾.
Equivalent fractions: Fractions with the same value, e.g. 1/2 = 2/4.
Simplest form: A fraction whose top and bottom have no common factor except 1.
Factor: A number that divides into another exactly.
HCF: Highest common factor: the biggest factor two numbers share.
Multiple: A number in a times table: 4, 8, 12, … are multiples of 4.
LCM: Lowest common multiple: the smallest multiple two numbers share.

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