Percentages and Decimals
A percentage is a number of hundredths, so it can also be written as a decimal. In NSW Year 7 maths (NESA outcome MA4‑FRC‑C‑01) you convert in both directions by moving the decimal point two places.
You will turn 0.253 into 25.3% and 61.3% into 0.613, handle fractional percentages such as \(2\frac{1}{2}\% = 0.025\), and fill empty places with zeros. You will then order mixed lists such as 0.4, 38%, 0.045 and 4.2%, and avoid the trap of thinking 0.5% equals 0.5. Interest rates, test results and sports statistics use both forms. This subtopic is part of Year 7 Percentages.
Theory
A percentage is a number of hundredths, so it can also be written as a decimal: \(36\% = 0.36\). To change a decimal to a percentage, multiply by \(100\); to change a percentage to a decimal, divide by \(100\). This Year 7 maths page shows the two-place move, fractional percentages, ordering and decimal multipliers.
A decimal is a fraction whose denominator is a power of \(10\): \(0.022 = \dfrac{22}{1000}\) (\(3\) decimal places, \(3\) zeros on the denominator). A percentage is a fraction out of \(100\): \(15\% = \dfrac{15}{100} = 0.15\).
Decimal to percentage: multiply by \(100\). The decimal point moves \(2\) places to the right: \(0.37 = 37\%\), \(0.126 = 12.6\%\).
Percentage to decimal: divide by \(100\). The decimal point moves \(2\) places to the left. Fill any empty places with zeros: \(41\% = 41 \div 100 = 0.41\), \(7.5\% = 0.075\).
A fractional percentage: change the fraction part to a decimal first: \(3\dfrac{1}{2}\% = 3.5\% = 0.035\).
\(100\% = 1\). More than \(1\) whole is more than \(100\%\): \(1.6 = 160\%\) and \(320\% = 3.2\).
Decimals worth knowing by heart:
| Decimal | Percentage |
|---|---|
| \(0.01\) | \(1\%\) |
| \(0.1\) | \(10\%\) |
| \(0.25\) | \(25\%\) |
| \(0.5\) | \(50\%\) |
| \(0.75\) | \(75\%\) |
| \(1\) | \(100\%\) |
| \(1.5\) | \(150\%\) |
How to change between decimals and percentages
- Decimal to percentage: \(\times\,100\), point \(2\) places right, add \(\%\).
- Percentage to decimal: \(\div\,100\), point \(2\) places left, remove \(\%\).
- Fill empty places with zeros.
- Check: a percentage less than \(100\%\) is a decimal less than \(1\).
(i) \(0.64\)
(ii) \(0.06\)
(iii) \(0.025\)
| \(0.64 \times 100\) | \(=\) | \(64\) |
| \(0.06 \times 100\) | \(=\) | \(6\) |
| \(0.025 \times 100\) | \(=\) | \(2.5\) |
(i) \(64\%\)
(ii) \(6\%\) (check: \(0.06 = \dfrac{6}{100}\))
(iii) \(2.5\%\)
(i) \(72\%\)
(ii) \(3\%\)
(iii) \(3\dfrac{1}{2}\%\)
| \(72 \div 100\) | \(=\) | \(0.72\) |
| \(3 \div 100\) | \(=\) | \(0.03\) |
| \(3.5 \div 100\) | \(=\) | \(0.035\) |
(i) \(0.72\)
(ii) \(0.03\)
(iii) \(\dfrac{1}{2} = 0.5\), so \(3\dfrac{1}{2}\% = 3.5\% = 0.035\)
(ii) Write \(0.004\) as a percentage.
(iii) Write \(410\%\) as a decimal.
| \(2.75 \times 100\) | \(=\) | \(275\) |
| \(0.004 \times 100\) | \(=\) | \(0.4\) |
| \(410 \div 100\) | \(=\) | \(4.1\) |
(i) \(275\%\)
(ii) \(0.4\%\)
(iii) \(4.1\)
| \(1.2 \times 100\) | \(=\) | \(120\) |
| \(0.09 \times 100\) | \(=\) | \(9\) |
\(9\% < 95\% < 110\% < 120\%\), so the order is \(0.09\), \(95\%\), \(110\%\), \(1.2\).
(i) a \(20\%\) discount?
(ii) a \(7\%\) rise?
(iii) Which change does \(\times\,1.3\) make?
| \(1 - 0.2\) | \(=\) | \(0.8\) |
| \(1 + 0.07\) | \(=\) | \(1.07\) |
(i) \(0.8\) (\(80\%\) is left). Check with \(\$50\): \(50 \times 0.8 = 40\) and \(50 - 10 = 40\).
(ii) \(1.07\) (\(107\%\))
(iii) \(1.3\) is \(130\%\): a \(30\%\) rise
| \(4.25 \div 100\) | \(=\) | \(0.0425\) |
\(0.0425\)
Common pitfalls
Frequently asked questions
How do you change a decimal to a percentage?
Multiply by \(100\): move the point \(2\) places right. \(0.37 = 37\%\).
How do you change a percentage to a decimal?
Divide by \(100\): move the point \(2\) places left. \(23\% = 0.23\).
What is 4% as a decimal?
\(0.04\).
What is 3 1/2% as a decimal?
\(3.5\% = 0.035\).
What decimal gives a 20% discount?
\(0.8\), because \(80\%\) of the price is left.