Lesson 4: Fractions, Decimals, and Percents

Three friends split a pizza so that everyone gets 3/4 of a small pizza each... that cannot be right — 3/4 is nearly a whole pizza. Fractions, decimals and percents are three languages for the same idea: a part of a whole. A recipe speaks fractions (3/4 cup of flour), a shop speaks percents (25% off), and a petrol pump speaks decimals (1.5 litres). In this lesson you will translate fluently between all three, compare values written in different forms, and find a percent of a quantity — skills you will use in every shop, kitchen and exam from now on.

The three forms of the same number

To turn a fraction into a decimal, divide the numerator by the denominator: 3/4 = 3 ÷ 4 = 0.75. To turn a decimal into a percent, multiply by 100 and add the % sign: 0.75 = 75%. To turn a percent into a fraction, write it over 100 and simplify: 75% = 75/100 = 3/4.

Some conversions deserve a permanent place in your memory:

The dots after 0.333… mean the 3s repeat forever. One third of anything never lands exactly on a decimal — a useful thing to know when a calculator shows 0.3333333.

Worked example 1. Write 5/8 as a decimal and a percent. Divide: 5 ÷ 8 = 0.625. Multiply by 100: 62.5%.

Worked example 2. Write 0.45 as a fraction in its simplest form. 0.45 = 45/100, and dividing top and bottom by 5 gives 9/20.

Comparing mixed forms

Which is bigger, 0.6 or 5/8? Convert to a common form. 5/8 = 0.625, and 0.625 > 0.6, so 5/8 is bigger. Decimals are usually the easiest meeting point: turn everything into a decimal and the order becomes obvious. The same trick settles arguments like "is 70% more than 2/3?" — 2/3 ≈ 0.667, so yes, 0.70 > 0.667, 70% wins.

Adding fractions with different denominators

To add fractions, the pieces must be the same size — you need a common denominator. For 1/4 + 2/3, the common denominator is 12: 1/4 = 3/12 and 2/3 = 8/12, so the sum is 11/12.

Worked example 3. A baker doubles a recipe that calls for 3/4 cup of flour: 2 × 3/4 = 6/4 = 1 1/2 cups. And if she first mixes 1/4 cup of cocoa with 2/3 cup of flour, the bowl holds 11/12 of a cup of dry ingredients, as computed above.

Finding a percent of a quantity

"Percent" means "out of 100", so 30% of 80 is 30/100 × 80 = 24. A flexible method: find 10% first (divide by 10), then build up. 10% of 80 is 8, so 30% is 3 × 8 = 24, and 5% (half of 10%) is 4. This "10% building block" method powers the whole of the next lesson on tax, tips and discounts.

Worked example 4. Find 15% of 60. 10% of 60 = 6; 5% = 3 (half of 6); 15% = 6 + 3 = 9.

In the real world: test scores and surveys

A test marked out of 40: scoring 34 gives 34/40 = 0.85 = 85%. News reports do the same with surveys: "3 out of 5 voters" becomes 3/5 = 0.6 = 60%. Battery icons, download progress bars and "chance of rain" forecasts are all percents wearing different clothes — and now you can translate every one of them back into fractions and decimals.

Common mistakes

• Adding numerators and denominators. 1/4 + 2/3 is not 3/7. Find a common denominator first: 3/12 + 8/12 = 11/12.
• Moving the decimal point the wrong way. To convert 0.625 to a percent, multiply by 100: 62.5%, not 6.25%. Percent numbers are bigger than their decimal twins.
• Thinking 0.5 is less than 1/3 because "5 is small". Convert: 1/3 ≈ 0.333, and 0.5 > 0.333, so 0.5 is bigger.
• Simplifying only once. 45/100 simplifies to 9/20 — divide top and bottom by the same number (5) until no common factor remains.

Exam Experience

1. What is 5/8 written as a percent? A) 5.8% B) 58% C) 62.5% D) 85%

2. Which fraction is equal to 0.45? A) 4/5 B) 9/20 C) 45/1000 D) 1/45

3. What is 40% of 50? A) 10 B) 20 C) 25 D) 40

4. (Constructed response) In a class survey, 18 out of 24 students said soccer was their favourite sport. (a) Write the fraction of students who chose soccer, in its simplest form. (b) Write that fraction as a decimal and as a percent. (c) The school has 800 students. If the class is typical of the whole school, predict how many students in the school would choose soccer.

Answer key and explanations

• 1. C — 5 ÷ 8 = 0.625, and 0.625 × 100 = 62.5%.
• 2. B — 0.45 = 45/100 = 9/20 after dividing top and bottom by 5.
• 3. B — 10% of 50 is 5, so 40% is 4 × 5 = 20.
• 4. (a) 18/24 = 3/4. (b) 3/4 = 0.75 = 75%. (c) 75% of 800 = 600 students.

Scoring guide for question 4 (3 points)

• 1 point for 3/4, correctly simplified from 18/24 (dividing by 6).
• 1 point for both conversions correct: 0.75 and 75%.
• 1 point for the prediction, 600, from a correct percent-of-quantity calculation (0.75 × 800 or equivalent).

Going further: fluency under pressure

The test of this lesson is speed with accuracy: converting, ordering and computing without hesitating. These examples mix the skills deliberately, the way real problems do.

Worked example 5. Put in order: 7/10, 0.72, 71%. As decimals: 0.70, 0.72, 0.71. Order: 7/10 < 71% < 0.72. Three different costumes, one number line.

Worked example 6. Two friends eat pizza: you eat 3/8 of it, your friend eats 1/4. Together: 3/8 + 2/8 = 5/8 eaten, so 1 − 5/8 = 3/8 remains. Fractions of a whole must always sum sensibly — if your parts total more than 1, something has gone wrong.

Worked example 7. Find 60% of 45 using blocks: 10% of 45 = 4.5, so 60% = 6 × 4.5 = 27. Decimal tenths inside percent work are normal — do not panic when 10% is not whole.

Worked example 8. Express 45 minutes as a fraction of an hour: 45/60, which simplifies to 3/4 of an hour, i.e. 0.75 hours. Timetables mix minutes and hours constantly, and this conversion (divide by 60, simplify) is the bridge.

Quick practice

• 1. Write 7/20 as a percent.
• 2. Write 0.08 as a fraction in simplest form.
• 3. Find 12% of 300.
• 4. Which is bigger, 0.35 or 1/3? By how much, roughly?

Answers. 1. 7/20 = 35/100 = 35%. 2. 8/100 = 2/25. 3. 10% = 30, 2% = 6, total 36. 4. 0.35 is bigger: 1/3 ≈ 0.333, so 0.35 wins by about 0.017.

Investigation: one budget, three languages

You have $50 to spend on ingredients, and three prices are quoted in three different languages. Item A costs 3/5 of your $50: 3/5 × 50 = $30. Item B costs 0.45 of $50: $22.50. Item C costs 70% of $50: $35. Before adding, notice the trap: buying all three costs 30 + 22.50 + 35 = $87.50, which exceeds the $50 budget — because each price was a fraction of the same $50, the three fractions (0.6 + 0.45 + 0.7 = 1.75) sum to more than one whole budget. Fractions of the same whole cannot be spent twice; that single insight prevents a lifetime of budget errors. Now a conversion workout: a quiz score of 27 out of 30 is 27/30 = 9/10 = 0.9 = 90%. And on the clock: 20 minutes is 20/60 = 1/3 of an hour, which is why a journey of "2 and 1/3 hours" is 2 hours 20 minutes, not 2 hours 30. Three languages, one quantity — translate early, translate everything to the same form, and only then compute.

Challenge corner

A class has 30 students, and exactly 2/3 of them are girls. That is 2/3 × 30 = 20 girls, and as a percent of the class, 20/30 = 2/3 ≈ 66.7%. The boys number 10, which is 1/3 ≈ 33.3% — and 66.7 + 33.3 = 100, as complementary parts of one whole must. Watch the rounding, though: 1/3 is 33.333…%, so reporting "33%" for each of three equal groups sums to 99%, a phantom missing percent that is purely a rounding artifact. News graphics fall into this trap weekly. Finish with a conversion sprint: 17/25 as a percent. Scale the denominator to 100 (multiply top and bottom by 4): 68/100 = 68%. Denominators of 20, 25 and 50 are friendly for exactly this reason — scale them to 100 and the percent reads itself off.