7.3 Percentages & Fraction–Decimal Equivalents

Fractions, decimals and percentages describe the same proportion in different ways. The Year 5 foundation is understanding hundredths and recognising equivalent representations. This section starts with those skills, then offers extension examples involving percentages of amounts and percentage changes. Simple scaling and optional formal ratio sharing are taught in the next chapter.

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1. Percentages and equivalent representations

The term percent originates from the Latin per centum, meaning "out of 100" or "per hundred". It is denoted by the symbol %. A percentage is simply a fraction with a fixed denominator of $100$:

$$P\% = \frac{P}{100} = P \times 0.01$$

Useful equivalences

Start with halves, quarters, fifths and tenths. The eighths and recurring thirds below are further connections to explore once the foundations are secure; memorising every row is not an official ISEB requirement:

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2. Build an equivalent fraction out of one hundred

Imagine a grid of one hundred equal squares. If seven squares are shaded, the shaded part is seven hundredths, written as 0.07 or 7%. If seventy squares are shaded, the part is seventy hundredths, written as 0.70 or 70%. The zero immediately after the decimal point matters: seven hundredths is much smaller than seven tenths. Adding a zero at the end of 0.7 gives 0.70 and leaves its value unchanged.

To convert a fraction, ask how its denominator can become one hundred. Multiply the numerator by the same factor so the proportion stays unchanged. For seven twenty-fifths, multiply both numbers by four: seven becomes twenty-eight, and twenty-five becomes one hundred. Therefore, seven twenty-fifths is twenty-eight hundredths, or 0.28 and 28%. Multiplying only the denominator would change the fraction rather than rename it.

For eighteen fiftieths, multiply both numbers by two to obtain thirty-six hundredths. This is 0.36 or 36%. Check by simplifying the original fraction to nine twenty-fifths, then multiplying both parts by four. The two routes agree. You do not need a memorised table entry for every numerator.

The same reasoning works in reverse. A label showing 45% means forty-five hundredths, so its decimal is 0.45. As a fraction, forty-five hundredths simplifies to nine twentieths after dividing both parts by five. It does not become forty-five tenths or 4.5, because those values are greater than one whole.

Compare using one representation

Order these proportions from smallest to largest: two fifths, 0.45 and 48%. Convert two fifths to forty hundredths, or 40%, and convert 0.45 to 45%. Now compare 40%, 45% and 48%. The original order is two fifths, then 0.45, then 48%. Comparing the visible numbers two, forty-five and forty-eight without accounting for their forms would not compare their values.

Find what remains

A box has twenty equal-sized counters, of which seven are blue. The blue fraction is seven twentieths. Multiply both parts by five to obtain thirty-five hundredths, so 35% are blue. Thirteen counters remain: thirteen twentieths is sixty-five hundredths, or 65%. The two percentages sum to 100%, just as the two counter counts sum to twenty. If asked for the percentage that is not blue, give 65%, rather than the percentage you found first. Always name the whole before calculating a part.

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3. Extension: calculating percentages of amounts

Year 5 introduces percentage equivalents and connects them to fractions of quantities. General percentage calculations develop further in Year 6. For these extension problems, splitting a percentage into familiar parts can make the calculation easier to check.

Useful parts of an amount

• To find $10\%$: Divide the amount by $10$ (shift all digits one place to the right).
• To find $1\%$: Divide the amount by $100$ (shift all digits two places to the right).
• To find $5\%$: Halve $10\%$.
• To find $20\%$: Double $10\%$.
• To find $25\%$: Divide by $4$ (or halve $50\%$).
• To find $50\%$: Divide by $2$.

Deconstructing Composite Percentages

Decomposition Strategy for 35% of £480:
  10% of £480 = £48
  30% = 3 × £48 = £144
   5% = half of £48 = £24
  35% = £144 + £24 = £168

Worked Example 1: Calculate $45\%$ of £360.

• Method: $50\% - 5\%$.
• $50\% \text{ of } 360 = 360 \div 2 = £180$.
• $5\% \text{ of } 360 = 180 \div 10 = £18$.
• $45\% = 180 - 18 = £162$.

Worked Example 2: Calculate $17.5\%$ of £400.

• Deconstruct into: $10\% + 5\% + 2.5\%$.
• $10\% \text{ of } 400 = £40$.
• $5\% = £40 \div 2 = £20$.
• $2.5\% = £20 \div 2 = £10$.
• Total: $40 + 20 + 10 = £70$.

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4. Extension: percentage increase and decrease

Word problems frequently ask candidates to adjust a price, measurement, or population by a given percentage.

The Two-Step Calculation Model

1. Step 1: Calculate the monetary or numerical value of the percentage change.
2. Step 2:
   • For an increase: Add the change to the original amount.
   • For a decrease: Subtract the change from the original amount.

Worked Example (Percentage Decrease): A jacket originally priced at £85 is placed on sale with a $20\%$ discount. What is the sale price of the jacket?

• Calculate $20\%$ of £85:
  • $10\% \text{ of } 85 = £8.50$.
  • $20\% = 8.50 \times 2 = £17.00$.
• Subtract discount from original price: $$\text{Sale Price} = 85 - 17 = £68.00$$

Worked Example (Percentage Increase): A village population of 2,400 residents increases by $15\%$ over a decade. What is the new population?

• Calculate $15\%$ of 2,400:
  • $10\% \text{ of } 2,400 = 240$.
  • $5\% \text{ of } 2,400 = 120$.
  • $15\% = 240 + 120 = 360$.
• Add increase to original population: $$\text{New Population} = 2,400 + 360 = 2,760$$

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