0580 · 1.4Fractions, decimals and percentages

Fraction forms and equivalence

0:00A phone battery showing seventy five percent has 3/4 of its full charge remaining. We could also describe that proportion as zero point seven five. These expressions look different, but they represent the same value. Understanding that link makes conversions easier, even for decimals that repeat forever. Fractions count equal parts, decimals use place value, and percentages give amounts per hundred. Each form offers its own way to express a number.

0:25By the end, you will also see why recurring decimals can be written as exact fractions. In three fifths, the denominator is five, showing how many equal parts make one whole. The numerator is three, showing how many of those parts we count. A fraction also represents division, so this fraction means 3 ÷ 5. A proper fraction has a numerator smaller than its denominator. A positive proper fraction is therefore worth less than one.

0:54An improper fraction has a numerator greater than or equal to its denominator. Its value is at least one, so 7/5 and 5/5 are both improper. A mixed number combines an integer with a proper fraction. One and two fifths means a complete whole plus two fifths. Each whole contains five fifths, so that amount also equals 7/5. For our first worked example, write two and three fifths as an improper fraction.

1:202 × 5 gives ten fifths in the whole number part. 10 + 3 gives thirteen fifths altogether. Therefore, two and three fifths equals 13/5. To reverse the conversion, divide thirteen by five. The result is two with a remainder of three. The two represents complete wholes, while the remaining three parts are still fifths. So 13/5 becomes two and three fifths, with the denominator unchanged. Equivalent fractions have the same value, even though their numerators and denominators differ.

1:52Splitting every fifth into two equal pieces turns three fifths into six tenths. We now count twice as many pieces, but each one is half as large. Multiplying both numerator and denominator by the same nonzero number preserves the value. For our second example, write 18/24 in its simplest form. The highest common factor of eighteen and twenty four is six. 18 ÷ 6 = 3. 24 ÷ 6 = 4.

2:19Therefore, 18/24 simplifies to 3/4. A fraction is in its simplest form when its numerator and denominator have no common factor greater than one. Dividing only the numerator would alter the fraction instead. Simplifying changes the appearance of a fraction, not the amount it stands for.

Converting decimals and percentages

2:38Decimal place value gives another way to describe equal parts. Take the decimal zero point three six. It means thirty six hundredths, so it equals 36/100. Dividing both numbers by four gives 9/25 in its simplest form. The number of decimal places tells us which denominator to use before simplifying. For our third example, write zero point three seven five as a fraction in its simplest form. Three decimal places mean thousandths, so write 375/1000.

3:08The highest common factor of these numbers is one hundred and twenty five. 375 ÷ 125 = 3. 1000 ÷ 125 = 8, so the answer is three eighths. To convert a fraction into a decimal, divide its numerator by its denominator. For 3/4, we can instead create a denominator of one hundred. Multiplying both parts by twenty five gives 75/100. Seventy five hundredths is zero point seven five, which also explains our battery example.

3:39Percent means per hundred, so seventy five percent is 75/100. To express a decimal as a percentage, multiply it by one hundred. 0.75 × 100 gives seventy five, so we write seventy five percent. To convert back, divide the percentage number by one hundred. To turn a percentage into a fraction, write it over one hundred and simplify, so forty percent becomes two fifths. For our fourth example, write seven twentieths as a decimal and a percentage.

4:09Multiplying its numerator and denominator by five gives 35/100. Thirty five hundredths equals zero point three five. Multiplying zero point three five by one hundred gives thirty five, so the percentage is thirty five percent. Percentages do not have to be whole numbers or stay below one hundred. Twelve point five percent equals zero point one two five, which is one eighth. One hundred and twenty five percent equals one point two five, or 5/4.

4:37Anything above one hundred percent simply represents more than one whole.

Recurring decimal notation and division

4:42Some decimals terminate, meaning their digits come to an end. Others recur, meaning a digit or block repeats without stopping. A recurring decimal is exact once its repeating pattern is marked. Writing only a few digits without a recurring mark gives a rounded value instead. In zero point one seven, a single dot above the seven means only that digit repeats. In zero point one two three, dots above the two and the three mark the first and last digits of the repeating block.

5:08Here the block two three repeats after the initial one. A bar over every digit of zero point one two three means the block one two three repeats as a whole. Always identify exactly where the repeating block begins and ends. For our fifth example, convert 8/45 into a decimal using division. Eight is smaller than forty five, so the whole number part is zero. 80 ÷ 45 gives one with a remainder of thirty five.

5:35This places one in the tenths position. For the next digit, 350 ÷ 45 gives seven with a remainder of thirty five. Repeating that step gives another seven and the same remainder. Because the remainder repeats, the following calculation must repeat too. Therefore, 8/45 = 0.1 followed by sevens recurring.

Recurring decimals as fractions

5:57We can reverse this process using a little algebra. Multiplying a decimal by ten shifts its digits one place to the left. We choose multipliers that place matching repeating tails beneath each other. Subtracting then removes those identical tails, leaving an equation involving whole numbers. For our sixth example, write zero point one seven, with only the seven recurring, as a fraction in its simplest form. Let x equal zero point one followed by sevens recurring.

6:24Multiplying by ten gives 10x = 1.7 recurring. Multiplying the original value by one hundred gives 100x = 17.7 recurring. Subtract 10x from 100x to obtain 90x. Subtract one point seven recurring from seventeen point seven recurring to obtain sixteen. Therefore, 90x = 16. Dividing by ninety gives x = 16/90. Simplifying by two gives the exact answer, 8/45. For our seventh example, write zero point one two three, with dots above the two and the three, as a fraction.

7:00Let x equal zero point one two three two three, continuing forever. Multiplying by ten moves the single non-recurring digit in front of the point. So 10x = 1.23, with the block two three recurring. Multiplying by one thousand gives 1000x = 123.23, with the same block recurring. Subtracting 10x from 1000x leaves 990x. The recurring tails cancel, so the right side becomes 123 − 1. That gives 990x = 122.

7:34Dividing gives x = 122/990. Halving both numbers gives the simplest form, 61/495. For our eighth example, let x equal zero point one two three, with a bar over all three digits. The block starts straight after the point, so 1000x = 123.123 recurring. Subtracting x leaves 999x = 123. So x = 123/999. Dividing both numbers by three gives 41/333. A common mistake is treating a recurring decimal as though it terminates.

8:15Zero point one seven is seventeen hundredths, but zero point one with the seven recurring is larger. Another error is putting the dots over the one and the three, instead of the two and the three. Students also forget to simplify, leaving answers such as 16/90. Check the dot positions and the simplest form before moving on.

Exam tips and recap

8:33Paper two is non-calculator, so these conversions must work by hand. Paper four permits a calculator, but showing the division, equivalent fractions or algebra still earns method marks. Exact answers such as 8/45 should stay exact rather than being rounded. If a question asks for an approximation, give three significant figures unless told otherwise. Never round intermediate values in the middle of a calculation. Fractions rest on equal parts, decimals on place value, and percentages on amounts per hundred.

9:03Equivalent fractions keep their value because numerator and denominator change together. Recurring decimals become exact fractions once matching repeating tails are removed by subtraction.