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E1.4 Fractions, decimals and percentages

Syllabus
0580–2028–2029
Topic
E1.4
Level
Extended

Learning objectives

E1.4.1Use the language and notation of the following in appropriate contexts:• Use the language and notation of the following in appropriate contexts: • proper fractions • improper fractions • mixed numbers • decimals • percentages.E1.4.2Recognise equivalence and convert between these forms• Recognise equivalence and convert between these forms. Candidates are expected to be able to write fractions in their simplest form. Recurring decimal notation is required, e.g. •..01 70 1777f=o •..0 123 0 1232323f=oo •..0 123 0 123123f= Includes converting between recurring decimals and fractions and vice versa, e.g. write.01 7o as a fraction.

Interpret fractions, decimals and percentages

Fractions, decimals and percentages are different ways to describe a quantity relative to one whole. The appropriate form depends on what needs to be clear: equal parts, decimal place value, or comparison with 100.

Form Meaning Example
proper fraction numerator is smaller than denominator, so the value is less than 1 35\frac35
improper fraction numerator is at least the denominator, so the value is at least 1 74\frac74
mixed number a whole-number part plus a proper fraction 134=1+341\frac34=1+\frac34
decimal digits after the point represent tenths, hundredths, and so on 0.250.25 is 25 hundredths
percentage a number of parts per 100 25%=2525\%=25 out of 100

In ab\frac ab, the denominator bb tells how many equal parts make one whole and the numerator aa tells how many of those parts are taken. The denominator cannot be zero.

The whole and the units matter. For example, 24.60 dollars as a fraction of 2870 dollars is 24.602870\frac{24.60}{2870} because both amounts use the same unit. A percentage can be greater than 100%100\% when a quantity exceeds the chosen whole.

A mixed number is a sum, so 2132\frac13 means 2+132+\frac13, not 2×132\times\frac13. A decimal is not automatically a percentage: 0.40.4 means four tenths, while 0.4%0.4\% means four tenths of one percent.

Convert equivalent forms, including recurring decimals

Equivalent fractions, decimals and percentages name the same value. A conversion changes the notation, not the quantity; finish fractional answers in simplest form.

Conversion Method Example
fraction →\to decimal numerator ÷\div denominator 38=0.375\frac38=0.375
decimal →\to fraction use place value, then simplify 0.24=24100=6250.24=\frac{24}{100}=\frac6{25}
decimal →\to percentage multiply by 100 and add %\% 0.375=37.5%0.375=37.5\%
percentage →\to decimal divide by 100 62%=0.6262\%=0.62
improper ↔\leftrightarrow mixed divide, or combine wholes and parts 74=134\frac74=1\frac34

Dots identify the recurring block. For example, 0.17˙=0.1777…0.1\dot7=0.1777\ldots, 0.12˙3˙=0.1232323…0.1\dot2\dot3=0.1232323\ldots, and 0.1˙23˙=0.123123…0.\dot1 2\dot3=0.123123\ldots. To convert a recurring decimal to a fraction, multiply by powers of 10 until two versions have the same recurring tail, then subtract so the tail cancels.

Let x=0.37‾=0.373737…x=0.\overline{37}=0.373737\ldots. Then 100x=37.373737…100x=37.373737\ldots. Subtracting gives 99x=3799x=37, so x=3799x=\frac{37}{99}.

For a non-recurring prefix, let x=0.419‾=0.4191919…x=0.4\overline{19}=0.4191919\ldots. Then 1000x=419.1919…1000x=419.1919\ldots and 10x=4.1919…10x=4.1919\ldots. Subtract: 990x=415990x=415, so x=415990=83198x=\frac{415}{990}=\frac{83}{198}.

To convert a fraction to a decimal, divide. If the digits repeat, mark the shortest complete recurring block: 211=0.181818…=0.18‾\frac2{11}=0.181818\ldots=0.\overline{18}. Check any conversion by evaluating the final fraction as a decimal.

Multiply both versions of xx by powers of 10 that align the same recurring tail; subtracting mismatched tails gives a false fraction. Dots belong over the first and last digits of the repeating block, not over a non-recurring prefix.