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Mathematics notes, unit 3

Fractions, Decimals and Percentages

Textbook Mathematics Year 3 (DLP), Part 1 and Part 2, pages Part 1, 81–114

What this chapter covers

  • You can name proper fractions and say which part is the numerator and which is the denominator (3.1.1)
  • You can find equivalent fractions and write a fraction in its simplest form (3.1.2, 3.1.3)
  • You can tell mixed numbers and improper fractions apart and change one into the other (3.1.7)
  • You can add and subtract fractions, including when one denominator is a multiple of the other (3.1.5, 3.1.6)
  • You can read, write and compare decimals up to hundredths (3.1.4, 3.2.1–3.2.3)
  • You can add and subtract decimals by lining up the decimal points (3.2.4, 3.2.5)
  • You can change between fractions of hundredths, decimals and percentages (3.3, 3.4)
  • You can solve everyday problems and create stories with fractions, decimals and percentages (3.5.1, 3.5.2)

Fractions, decimals and percentages are everywhere. A shop sign says 50% off. Your pen tip is 0.7 mm. Mak cuts a cake into equal pieces and gives you 14. In this unit you will learn to read, write, add and subtract them.

3.1Proper fractions

A fraction tells us how many equal parts of a whole we have. The top number is the numerator. The bottom number is the denominator.
A circle cut into 4 equal parts, 3 parts red
3 of 4 equal parts are red. We say "three over four".
PartWhat it tells usIn 34
Numerator (top)How many parts we take3
Denominator (bottom)How many equal parts in the whole4

34 is a proper fraction. In a proper fraction the numerator is smaller than the denominator, so its value is less than 1.

We can also find fractions of a group of things. Count all the things. That is the denominator.

8 balloons: 3 red, 2 blue, 2 yellow, 1 purple
Balloons at a birthday party
What fraction of the balloons are red? What fraction are purple?
Show answerHide answer
There are 8 balloons. Red: 38 (three over eight). Purple: 18.
The parts must be equal. A roti canai torn into 4 pieces of different sizes does not show quarters.

3.2Equivalent fractions and the simplest form

Equivalent fractions are different fractions that have the same value. "Equivalent" means "equal in value".
Two bars of the same length: 2 of 3 parts shaded, and 4 of 6 parts shaded
The blue parts are the same size, so 23 = 46.

Use the fraction chart. Look at the red dashed line. In the rows for halves, quarters, sixths, eighths and tenths, the parts fit exactly up to the line, so each of these shows a fraction equal to 12. In the rows for thirds and fifths, the line cuts through the middle of a part, so no number of thirds or fifths is equal to 12.

Fraction chart with rows for 1, halves, thirds, quarters, fifths, sixths, eighths and tenths
Fraction chart: 12 = 24 = 36 = 48 = 510

Example 1: Find an equivalent fraction

23 = ?6

  1. 3 becomes 6. We multiplied by 2.
  2. Do the same to the top: 2 × 2 = 4.
  3. So 23 = 46.
To find an equivalent fraction, multiply or divide the numerator and the denominator by the same number.

Simplest form

A fraction is in its simplest form when the numerator and the denominator can only both be divided by 1.

Example 2: Simplify 410

  1. 4 and 10 can both be divided by 2.
  2. 4 ÷ 2 = 2 and 10 ÷ 2 = 5.
  3. 2 and 5 cannot both be divided again. So 410 = 25.
4 of 10 parts shaded is the same length as 2 of 5 parts shaded
Divide both numbers. 68 is not 38. 6 ÷ 2 = 3 and 8 ÷ 2 = 4, so 68 = 34.
Write 39 and 610 in the simplest form.
Show answerHide answer
39: divide by 3 → 13. 610: divide by 2 → 35.

3.3Mixed numbers and improper fractions

One whole pie and one quarter of a pie
One whole pie and 14 of a pie: 114 pies
A mixed number has a whole number and a proper fraction, like 114 ("one and one over four"). Its value is more than 1.
An improper fraction has a numerator that is larger than or equal to the denominator, like 116 or 66.

Example 3: Two hexagons

Two hexagons each cut into 6 parts; 11 parts are green
  1. Each hexagon has 6 equal parts, so each part is 16.
  2. Count the green parts: 6 + 5 = 11.
  3. As an improper fraction: 116. As a mixed number: 156.
229 = 209 because 2 wholes = 189, and 189 + 29 = 209.

3.4Adding and subtracting fractions

Same denominator

When the denominators are the same, add or subtract the numerators only. Keep the denominator.

Number line in ninths: jump of 2 ninths then 5 ninths, landing on 7 ninths
29 + 59 = 79
Number line in sevenths: jump back 4 sevenths from 6 sevenths to 2 sevenths
67 − 47 = 27
Do not add the denominators! 14 + 24 is not 38. Quarters plus quarters are still quarters: 34.

Different denominators

First make the denominators the same. Use an equivalent fraction.

Example 4: 12 + 14

  1. Change halves into quarters: 12 = 24.
  2. Add: 24 + 14 = 34.
Bars showing 1/2 equals 2/4, plus 1/4, gives 3/4

Example 5: 56 − 12

  1. Change halves into sixths. Multiply the top and bottom by 3: 12 = 36.
  2. Subtract: 56 − 36 = 26.
  3. Simplify: 26 = 13.
Always write the answer in its simplest form. 78 − 38 = 48 = 12.
a) 35 + 310 = ?
b) 910 − 25 = ?
Show answerHide answer
a) 35 = 610, so 610 + 310 = 910.
b) 25 = 410, so 910 − 410 = 510 = 12.

3.5Decimals

The kitchen floor has 100 tiles. 16 tiles are blue.

Hundred square with 16 squares shaded blue
16 of 100 is 16100 (sixteen hundredths)
A decimal is another way to write tenths and hundredths. 16100 = 0.16. We read it as "zero point one six". The dot is the decimal point.
FractionOnes.TenthsHundredthsDecimalRead as
161000.160.16zero point one six
41000.040.04zero point zero four
3100.30.3zero point three
Say each digit after the point. 0.72 is "zero point seven two", not "zero point seventy-two". And 4100 is 0.04, not 0.4.

Comparing decimals

Which is heavier, 0.45 kg or 0.2 kg? Line up the decimal points. Compare the tenths first: 4 tenths is more than 2 tenths. So 0.45 kg is heavier.

Decimal number line from 0.00 to 0.12 with 0.03 and 0.07 highlighted
0.03 comes before 0.07, so 0.03 is smaller.
A longer decimal is not always bigger. 0.4 = 0.40, which is larger than 0.38.

3.6Adding and subtracting decimals

Add and subtract decimals just like whole numbers. The big rule: line up the decimal points.

Bars of tenths: 3 tenths plus 5 tenths make 8 tenths
A shelf has parts 0.3 m and 0.5 m high: 0.3 m + 0.5 m = 0.8 m

Example 6: 0.25 ℓ + 0.4 ℓ

  1. Write 0.4 as 0.40 so both have hundredths.
  2. Line up the points: 0.25 + 0.40.
  3. Add hundredths: 5 + 0 = 5. Add tenths: 2 + 4 = 6.
  4. Answer: 0.65 ℓ.

Example 7: 0.6 kg − 0.13 kg

  1. Write 0.6 as 0.60.
  2. We cannot take 3 hundredths from 0 hundredths. Regroup 1 tenth into 10 hundredths: 0.60 becomes 5 tenths and 10 hundredths.
  3. 10 − 3 = 7 hundredths. 5 − 1 = 4 tenths.
  4. Answer: 0.47 kg. Check: 0.47 + 0.13 = 0.60 ✓
a) 0.45 + 0.36 = ?
b) 0.87 − 0.09 = ?
Show answerHide answer
a) 0.45 + 0.36 = 0.81 (5 + 6 = 11 hundredths, carry 1 tenth).
b) 0.87 − 0.09 = 0.78 (regroup: 17 − 9 = 8 hundredths).

3.7Percentages

Percent means "out of 100". The symbol is %. 26100 = 26% (twenty-six percent).
Hundred square with 26 squares shaded pink
26 of 100 squares are pink: 26%

A fraction of hundredths, a decimal and a percentage can show the same amount:

FractionDecimalPercentage
421000.4242%
81000.088%
691000.6969%
951000.9595%
8% is 0.08, not 0.8. 8 hundredths needs a 0 in the tenths place.
Did you know? More than 60% of your body is water. A jellyfish is about 95% water!

3.8Solving problems and creating stories

Use four steps: Given (what you know), Find (what is asked), Method (number sentence) and Check.

Example 8: At the pasar malam

Aisyah buys 0.9 kg of prawns and 0.55 kg of squid. How much more prawns than squid does she buy?

Bar model: prawns 0.9 kg, squid 0.55 kg, difference marked with a question mark
  1. Given: prawns 0.9 kg, squid 0.55 kg. Find: the difference.
  2. Method: 0.90 − 0.55 = 0.35
  3. Check: 0.35 + 0.55 = 0.90 ✓
  4. Answer: She buys 0.35 kg more prawns.

Example 9: Kuih for the kenduri

Mak Long uses 12 kg of flour and 14 kg of sugar. What is the total mass?

  1. 12 kg = 24 kg
  2. 24 kg + 14 kg = 34 kg
  3. The total mass is 34 kg.

Example 10: Chess Club

There are 100 pupils in the Chess Club. 45 are girls. What percentage are boys?

  1. Boys: 100 − 45 = 55
  2. 55 of 100 = 55100 = 55%

Create a story

You can turn a number sentence into a story. Use words like altogether and total for +, and left, difference and how much more for −.

Story for 0.6 ℓ − 0.25 ℓ = 0.35 ℓ

A bottle has 0.6 ℓ of mineral water. Ali drinks 0.25 ℓ after football. There is 0.35 ℓ of water left.

At the school canteen, 14 of a tray of nasi lemak is sold at recess and 38 after school. What fraction of the tray is sold altogether?
Show answerHide answer
14 = 28. 28 + 38 = 58 of the tray.

Summary

  • Fraction: numerator (parts taken) over denominator (equal parts in the whole). A proper fraction is less than 1.
  • Equivalent fractions have the same value. Multiply or divide the top and bottom by the same number.
  • Simplest form: divide top and bottom until only 1 divides both.
  • Mixed number = whole number + proper fraction, e.g. 114. Improper fraction: numerator ≥ denominator, e.g. 54.
  • Add or subtract fractions: make the denominators the same, then work with the numerators only. Simplify the answer.
  • Decimals: 310 = 0.3, 16100 = 0.16. Line up the decimal points to add and subtract.
  • Percent means out of 100: 42100 = 0.42 = 42%.