Fractions as Percentages

Imagine two paint mixtures that look almost the same shade of orange. One contains three cups of red paint in every four cups of mixture. Another contains seven cups of red paint in every ten cups of mixture. The numbers alone do not immediately tell you which mixture has the greater share of red. We need a way to describe both mixtures using the same size of reference group, while keeping each mixture’s proportions unchanged.

Percentages give us that shared reference. Instead of asking how much red there is in four cups or ten cups, we ask how much there would be in every hundred equal cups of mixture. We are changing the way the share is written; we are not adding paint or changing its colour.

You already know that one half can also be written as two quarters. A percentage is another way to write a fraction. The useful new idea is to make the denominator the same for every comparison.

Think of a square divided into one hundred equal small squares. If twenty-five are shaded, the shaded fraction is 25/100. Calling this 25% describes exactly the same shaded part. The small squares must be equal in area: counting unequal pieces would not correctly measure the fraction of the whole.

The hundred parts do not have to be separate physical objects. A ribbon, a journey, a mass of paint or a sum of money can all be imagined as one hundred equal parts. Thus 25% can describe one quarter of a whole even when the whole contains only four objects or measures less than one metre.

Expressing Fractions as Percentages

Start with the red paint fraction 3/4. If each of the four equal parts is divided into twenty-five smaller equal parts, the whole has 4 × 25 = 100 small parts. The three red parts then contain 3 × 25 = 75 small parts. Therefore 3/4 = 75/100 = 75%.

Notice why we multiply both the numerator and the denominator by twenty-five. Multiplying only the numerator would change the share. Equivalent fractions preserve the relationship between part and whole because both numbers are scaled by the same factor.

When the denominator divides one hundred easily, this method is convenient. For 2/5, multiply both numbers by twenty. For 9/20, multiply both by five. The new numerator tells you the percentage because the denominator is now one hundred.

What if a denominator does not divide one hundred into a whole number? The fraction 1/3 is a familiar example. Each third still represents one third of 100%, namely 33⅓%. A percentage may contain a fraction or a decimal; it does not have to be a whole number.

A general conversion follows from the same reasoning. If a fraction is a/b and its percentage number is x, then a/b = x/100. Multiplying both sides by 100 gives x = 100a/b. Remember that x is the number before the percent sign. The value of 75% is 0.75, not 75.

To travel in the opposite direction, read the percent sign as “divided by one hundred”. Thus 24% = 24/100. Then reduce the fraction by dividing numerator and denominator by a common factor. Dividing by four gives 6/25.

Simplifying is useful because a familiar fraction often reveals meaning. For example, 50/100 reduces to 1/2 and 75/100 to 3/4. Do not divide the numerator by one hundred twice: the percent sign has already supplied that division.

Percentages help us compare fractions with different denominators. Suppose sugar makes up 9/34 of one biscuit mixture and 13/45 of another. Their approximate percentages are 26.47% and 28.89%. The second mixture has the larger sugar proportion, even though simply comparing the numerators would not have been a reliable method.

Why use one hundred? Our decimal number system groups in tens, hundreds and thousands. A fraction out of one hundred is easy to connect with decimal hundredths. Other reference sizes are possible, but using the same reference consistently makes comparisons easier.

Percentages Around Us

A phone battery display, a progress bar and an ingredient label all use percentages to express a share relative to something. A battery showing 60% means sixty hundredths of the charge measure used by that device. A task showing 60% complete means sixty hundredths of its total work measure. The numbers look alike, but their wholes are different.

Whenever you read a percentage, finish the sentence “This is a percentage of …”. That habit will become especially useful when two packets have different masses or two journeys have different lengths. Equal percentages describe equal proportions; they do not necessarily describe equal quantities.