Fractions: types, equivalent fractions and simplifying

A fraction is one of the most fundamental ideas in mathematics. It tells you how many parts you have out of a whole. When Aminata buys a loaf of bread and cuts it into $4$ equal pieces, each piece is $\frac{1}{4}$ of the loaf. If she eats $3$ of those pieces, she has eaten $\frac{3}{4}$ of the loaf. Simple — but the trouble comes when fractions appear in WAEC questions with awkward numbers, mixed forms, or comparisons.

This lesson covers everything about fractions: their structure, the different types, equivalent fractions, comparing fractions, and simplifying them. The next lesson handles the four operations, conversion between fractions and decimals, and approximation. By the end of the two lessons you will be able to handle any WAEC fraction question with confidence.

Learning objectives

By the end of this lesson you should be able to:

• describe the parts of a fraction and the types of fraction
• convert between mixed numbers and improper fractions
• find equivalent fractions and simplify fractions
• compare and order fractions

The structure of a fraction

Every fraction has two numbers: the top (numerator) and the bottom (denominator), separated by a line called the fraction bar (which actually means divided by).

$$\frac{a}{b} = a \div b$$

So $\dfrac{3}{4}$ is the same as $3 \div 4 = 0.75$. We use this fact when we convert between fractions and decimals in the next lesson.

What the numbers mean

• Numerator ($a$): how many parts you are taking.
• Denominator ($b$): how many parts the whole is divided into.

If I divide a loaf of bread into $5$ equal pieces and you take $2$ of them, you have $\dfrac{2}{5}$ of the bread. The $5$ is the denominator (total slices), the $2$ is the numerator (slices you took).

Visualising fractions

Imagine a rectangular bar:

• Divided into $4$ equal parts, $3$ shaded → $\dfrac{3}{4}$
• Divided into $8$ equal parts, $5$ shaded → $\dfrac{5}{8}$
• Divided into $2$ equal parts, $1$ shaded → $\dfrac{1}{2}$

The denominator tells you how fine the divisions are. The numerator tells you how many of those divisions you are taking.

Types of fractions

Proper fractions

The numerator is less than the denominator. The value is less than $1$.

• $\dfrac{1}{2}, \dfrac{3}{4}, \dfrac{5}{8}, \dfrac{7}{10}$ are all proper fractions.

Improper fractions

The numerator is greater than or equal to the denominator. The value is $1$ or more.

• $\dfrac{5}{4}, \dfrac{7}{3}, \dfrac{8}{8}, \dfrac{15}{4}$ are all improper.

An improper fraction can be rewritten as a mixed number (next type).

Mixed numbers

A whole number followed by a proper fraction.

• $1\tfrac{1}{4}$ means one whole plus a quarter — equal to $\dfrac{5}{4}$.
• $2\tfrac{2}{3}$ means two wholes plus two thirds — equal to $\dfrac{8}{3}$.
• $3\tfrac{3}{4}$ means three and three quarters — equal to $\dfrac{15}{4}$.

Equivalent fractions

Different-looking fractions that represent the same value.

• $\dfrac{1}{2} = \dfrac{2}{4} = \dfrac{3}{6} = \dfrac{4}{8} = \dfrac{50}{100}$
• $\dfrac{3}{4} = \dfrac{6}{8} = \dfrac{9}{12} = \dfrac{12}{16}$

You get equivalent fractions by multiplying or dividing both numerator and denominator by the same nonzero number.

Converting between mixed numbers and improper fractions

Mixed → improper

Multiply the whole number by the denominator, add the numerator, keep the denominator.

Example. Convert $2\tfrac{3}{5}$ to improper.

Example. Convert $4\tfrac{2}{7}$ to improper.

Improper → mixed

Divide the numerator by the denominator. The quotient is the whole-number part; the remainder over the denominator is the fraction part.

Example. Convert $\dfrac{17}{4}$ to mixed.

$17 \div 4 = 4$ remainder $1$. So $\dfrac{17}{4} = 4\tfrac{1}{4}$.

Example. Convert $\dfrac{23}{5}$ to mixed.

$23 \div 5 = 4$ remainder $3$. So $\dfrac{23}{5} = 4\tfrac{3}{5}$.

Equivalent fractions in detail

Two fractions are equivalent if they have the same value, even if they look different. You produce equivalent fractions by scaling:

Examples.

• $\dfrac{2}{3} = \dfrac{2 \times 2}{3 \times 2} = \dfrac{4}{6}$ (multiplied top and bottom by $2$)
• $\dfrac{4}{6} = \dfrac{4 \div 2}{6 \div 2} = \dfrac{2}{3}$ (divided top and bottom by $2$)
• $\dfrac{1}{4} = \dfrac{5}{20}$ (both multiplied by $5$)
• $\dfrac{8}{12} = \dfrac{2}{3}$ (both divided by $4$)

The second form — dividing — is how we simplify fractions (next section).

Testing whether two fractions are equivalent

The cross-multiplication test: $\dfrac{a}{b} = \dfrac{c}{d}$ if and only if $a \times d = b \times c$.

Example. Is $\dfrac{3}{4}$ equivalent to $\dfrac{9}{12}$?

Cross-multiply: $3 \times 12 = 36$ and $4 \times 9 = 36$. Equal, so yes they are equivalent.

Example. Is $\dfrac{2}{3}$ equivalent to $\dfrac{6}{10}$?

Cross-multiply: $2 \times 10 = 20$ and $3 \times 6 = 18$. Not equal, so no they are not equivalent.

Simplifying fractions

A fraction is in simplest form (or lowest terms) when the numerator and denominator have no common factors other than $1$.

To simplify: divide both numerator and denominator by their greatest common factor (GCF, also called HCF — highest common factor).

Example. Simplify $\dfrac{8}{12}$.

Factors of $8$: $1, 2, 4, 8$. Factors of $12$: $1, 2, 3, 4, 6, 12$. Greatest common factor: $4$.

Example. Simplify $\dfrac{18}{24}$.

Factors of $18$: $1, 2, 3, 6, 9, 18$. Factors of $24$: $1, 2, 3, 4, 6, 8, 12, 24$. Greatest common factor: $6$.

$$\frac{18}{24} = \frac{3}{4}$$

Example. Simplify $\dfrac{45}{60}$.

GCF of $45$ and $60$ is $15$.

$$\frac{45}{60} = \frac{3}{4}$$

Step-by-step simplification

If you can't spot the GCF immediately, divide by smaller common factors repeatedly until no more divide both.

Example. Simplify $\dfrac{48}{72}$.

Step 1: Both even. Divide by $2$. $\dfrac{48}{72} = \dfrac{24}{36}$. Step 2: Both still even. Divide by $2$. $\dfrac{24}{36} = \dfrac{12}{18}$. Step 3: Both even. Divide by $2$. $\dfrac{12}{18} = \dfrac{6}{9}$. Step 4: Both divisible by $3$. Divide by $3$. $\dfrac{6}{9} = \dfrac{2}{3}$. Step 5: $2$ and $3$ have no common factor (other than $1$). Done.

Final: $\dfrac{48}{72} = \dfrac{2}{3}$.

Comparing fractions

Which is bigger, $\dfrac{3}{4}$ or $\dfrac{5}{8}$? You can't easily tell at a glance because the denominators are different. There are two reliable methods.

Method 1 — Common denominator

Convert both fractions to have the same denominator. Then compare the numerators.

Example. Compare $\dfrac{3}{4}$ and $\dfrac{5}{8}$.

The least common multiple (LCM) of $4$ and $8$ is $8$. Convert:

• $\dfrac{3}{4} = \dfrac{6}{8}$ (multiplied top and bottom by $2$)
• $\dfrac{5}{8}$ stays the same.

Now compare: $\dfrac{6}{8} > \dfrac{5}{8}$. So $\dfrac{3}{4} > \dfrac{5}{8}$.

Method 2 — Cross-multiplication

For $\dfrac{a}{b}$ vs $\dfrac{c}{d}$: compare $a \times d$ with $b \times c$.

• If $a \times d > b \times c$, then $\dfrac{a}{b} > \dfrac{c}{d}$.
• If $a \times d < b \times c$, then $\dfrac{a}{b} < \dfrac{c}{d}$.
• If $a \times d = b \times c$, they are equal.

Example. Compare $\dfrac{3}{7}$ and $\dfrac{4}{9}$.

$3 \times 9 = 27$. $7 \times 4 = 28$. Since $27 < 28$, we have $\dfrac{3}{7} < \dfrac{4}{9}$.

Ordering several fractions

To order several fractions, find a common denominator for all of them, then compare numerators.

Example. Arrange in ascending order: $\dfrac{2}{3}, \dfrac{3}{4}, \dfrac{5}{6}, \dfrac{7}{12}$.

LCM of $3, 4, 6, 12$ is $12$. Convert each:

• $\dfrac{2}{3} = \dfrac{8}{12}$
• $\dfrac{3}{4} = \dfrac{9}{12}$
• $\dfrac{5}{6} = \dfrac{10}{12}$
• $\dfrac{7}{12} = \dfrac{7}{12}$

Now order the numerators: $7 < 8 < 9 < 10$. So:

$$\frac{7}{12} < \frac{2}{3} < \frac{3}{4} < \frac{5}{6}$$

Fractions in real life — Sierra Leonean examples

Fractions appear everywhere. Some examples:

• A mother divides $\dfrac{2}{3}$ of a kilo of rice between her two children. Each gets $\dfrac{1}{3}$ kilo.
• A farmer in Bo harvests $\dfrac{4}{5}$ of his cassava crop in October, leaving $\dfrac{1}{5}$ for November.
• A school survey shows $\dfrac{3}{5}$ of students prefer English over Maths. If there are $200$ students, that's $\dfrac{3}{5} \times 200 = 120$ students.
• A market trader sells $\dfrac{7}{8}$ of her stock by lunchtime, leaving $\dfrac{1}{8}$ for the afternoon.

Understanding fractions makes all these calculations natural.

WASSCE examination focus

Practice problems

1. Simplify $\dfrac{24}{36}$.

2. Simplify $\dfrac{90}{135}$.

3. Convert $4\tfrac{2}{5}$ to an improper fraction.

4. Convert $\dfrac{29}{7}$ to a mixed number.

5. Compare: which is greater, $\dfrac{5}{8}$ or $\dfrac{7}{12}$?

6. Compare: $\dfrac{4}{9}$ vs $\dfrac{5}{11}$.

7. Are $\dfrac{8}{12}$ and $\dfrac{14}{21}$ equivalent?

8. Arrange in ascending order: $\dfrac{3}{4}, \dfrac{5}{6}, \dfrac{7}{12}, \dfrac{2}{3}$.

9. A bag contains $\dfrac{3}{5}$ kg of rice. If $\dfrac{1}{4}$ kg is used, what fraction of the original bag is left?

10. Find a fraction equivalent to $\dfrac{3}{8}$ with a denominator of $40$.

Worked solutions

1. GCF$(24, 36) = 12$. So $\dfrac{24}{36} = \dfrac{2}{3}$.

2. GCF$(90, 135) = 45$. So $\dfrac{90}{135} = \dfrac{2}{3}$.

3. $4\tfrac{2}{5} = \dfrac{(4 \times 5) + 2}{5} = \dfrac{22}{5}$.

4. $29 \div 7 = 4$ remainder $1$. So $\dfrac{29}{7} = 4\tfrac{1}{7}$.

5. Cross-multiply: $5 \times 12 = 60$; $8 \times 7 = 56$. $60 > 56$, so $\dfrac{5}{8} > \dfrac{7}{12}$.

6. Cross-multiply: $4 \times 11 = 44$; $9 \times 5 = 45$. $44 < 45$, so $\dfrac{4}{9} < \dfrac{5}{11}$.

7. Cross-multiply: $8 \times 21 = 168$; $12 \times 14 = 168$. Equal, so yes they are equivalent. (Both simplify to $\dfrac{2}{3}$.)

8. Common denominator: LCM$(4, 6, 12, 3) = 12$. Convert: $\dfrac{3}{4} = \dfrac{9}{12}$; $\dfrac{5}{6} = \dfrac{10}{12}$; $\dfrac{7}{12} = \dfrac{7}{12}$; $\dfrac{2}{3} = \dfrac{8}{12}$. Order: $7 < 8 < 9 < 10$. Answer: $\dfrac{7}{12} < \dfrac{2}{3} < \dfrac{3}{4} < \dfrac{5}{6}$.

9. This requires subtraction (covered fully in the next lesson). Common denominator: LCM$(5, 4) = 20$. $\dfrac{3}{5} = \dfrac{12}{20}$. $\dfrac{1}{4} = \dfrac{5}{20}$. Difference: $\dfrac{12 - 5}{20} = \dfrac{7}{20}$ kg left. As a fraction of the original $\dfrac{3}{5}$ kg bag: $\dfrac{7/20}{3/5} = \dfrac{7}{20} \times \dfrac{5}{3} = \dfrac{35}{60} = \dfrac{7}{12}$ of the bag is left.

10. $\dfrac{3}{8} = \dfrac{?}{40}$. Multiply both by $5$: $\dfrac{3 \times 5}{8 \times 5} = \dfrac{15}{40}$.

What's next

In the next lesson we cover operations with fractions (adding, subtracting, multiplying and dividing, including mixed numbers), converting between fractions and decimals, and approximation: decimal places, significant figures and estimation. These are the workhorse skills that appear in every WAEC paper, often in disguised form (recipe quantities, distance calculations, money problems).