Converting between fractions, decimals and percentages

Two shops are selling the same trainers. One window says three eighths off. The other says thirty five percent off. Same shoes, same price tag, two different signs. You cannot line those signs up in your head, because one speaks in fractions and the other speaks in percentages. Until they speak the same language, there is no way to tell where the bigger discount is. In this video you'll learn to move any value between fraction, decimal and percentage, and then use that skill to settle comparisons exactly like the trainers.

First, the method. We'll build it on one question: convert seven twentieths to a decimal and a percentage.

Before any working, the big idea. A fraction, a decimal and a percentage are not three different numbers. They are three costumes for the same value. Converting doesn't change the number, it just changes the costume.

Costume one to costume two. A fraction bar is a hidden divide sign: seven twentieths literally means seven divided by twenty. Do that division and you get nought point three five. Fraction to decimal: top divided by bottom.

Costume two to costume three. Percent means per hundred, out of one hundred. So a decimal becomes a percentage when you multiply by one hundred: nought point three five becomes thirty five percent. To come back the other way, divide by one hundred.

And that's the whole first answer. Seven twentieths equals nought point three five equals thirty five percent. One value, three costumes.

There's a second route worth knowing. Scale the fraction until the bottom is one hundred. Twenty times five is one hundred, and the top and the bottom travel together, so seven times five makes thirty five. Seven twentieths is thirty five out of one hundred: thirty five percent, no division needed.

Your turn, and it's a small step you can mark yourself. What is nought point six as a percentage: six percent, sixty percent, or nought point six percent?

Have a think. I'll wait.

The answer is sixty percent. Multiplying by one hundred slides the digits two places, so nought point six becomes sixty. If you picked six percent, you only slid one place. That's multiplying by ten, and it's the classic slip on this conversion.

One more tool completes the kit: decimal back to fraction. Read the place value. Nought point three five ends in the hundredths column, so it's thirty five over one hundred. Then cancel: divide the top and the bottom by five, and you're back to seven twentieths. For a refresher on cancelling, see the video on equivalent fractions.

Now the main event: using conversion to settle a comparison.

Here's the question. Which is bigger: seven sixteenths, or forty five percent? Show your working by converting both to the same form.

Pause and try it yourself. I'll wait.

Here's the working. Seven sixteenths: seven divided by sixteen gives nought point four three seven five. Multiply by one hundred: forty three point seven five percent. Now both values wear the identical costume. Forty three point seven five percent sits next to forty five percent, and forty five is bigger. So forty five percent is the larger value. Look at what made that conclusion safe. We didn't stop at one value as a decimal and the other as a percentage and call it close enough. Both values landed in exactly the same form, written side by side, before we compared. That final matching pair is the line that proves your answer.

Back to the trainers from the start. Three eighths is three divided by eight: nought point three seven five, so thirty seven point five percent. Put that next to thirty five percent, and the three eighths sign is the bigger discount. Same skill, real shop window.

Time for the examiner's-eye view: two habits that protect this skill under exam conditions.

Habit one: identical form, not just similar form. On a comparison question very like ours, OCR's examiners wrote: This was usually well answered. Candidates understood the need to compare the figures in a similar form and the most common conversion was five eighths to sixty two point five percent.

A similar form gets you moving. An identical form finishes the job. An answer that leaves one value as a decimal and the other as a percentage hasn't actually shown the comparison. Land both in the same form and write them next to each other.

Habit two: a rounded decimal is not the number you started with. Round seven sixteenths to one decimal place and you get nought point four. Compare that with nought point four three seven, and nought point four looks smaller. But the exact value, nought point four three seven five, is bigger. Rounding flipped the verdict.

The same OCR report describes exactly this habit: A significant number of candidates chose to change fractions to their decimal form in order to carry out processes (such as finding a fraction of an amount), rather than working with the fraction. Often they then used a rounded decimal and so lost accuracy.

So use conversion to compare and to present. If a calculation is still to come, like finding a fraction of an amount, stay with the exact fraction all the way through, and convert only at the very end, if at all. That's covered properly in the video on fraction of a quantity.

One reassurance before we leave the exam room. If a question simply says convert, any correct equivalent form earns the credit. Nought point three five and thirty five percent are both accepted forms of seven twentieths, so there's no need to hedge by writing every version.

Right. The whole toolkit, gathered on one screen.

Fraction to decimal: the bar is a divide sign, top divided by bottom. Decimal to percentage: multiply by one hundred. Percentage to decimal: divide by one hundred. Terminating decimal to fraction: read the place value, then cancel. And when you scale a bottom up to one hundred, the top and the bottom travel together. And the habit that holds it all together: one value, three costumes. Before you compare two values, dress them in the identical costume, and never trust a rounded decimal to do an exact job.

Next in the chapter: converting recurring decimals into fractions. A quick heads-up: that one is a Higher tier topic.

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