The FDP Trio — Fractions, Decimals, and Percentages

A bottle can be described as half full, 0.5 full or 50% full. These descriptions may look different, but they point to the same amount in the same bottle. Each description is useful in a different situation. A fraction helps you picture equal shares, a decimal makes many multiplications convenient, and a percentage places a share on the familiar scale of one hundred. Being able to move between them gives you several ways to think about one problem.

The letters FDP stand for fractions, decimals and percentages. They are not three separate kinds of quantity. They are three forms in which the same numerical value can be written. Changing form should never change the amount being described.

We will connect these forms carefully, then use them to interpret mixtures. Mixture problems require an extra decision: a ratio between two ingredients must first be related to the total mixture.

Consider 50%. By its meaning, this is 50/100. Simplifying gives 1/2, and dividing 1 by 2 gives 0.5. Therefore multiplying a whole by 0.5 has the same effect as taking half of it or finding 50% of it.

Similarly, 10% = 10/100 = 1/10 = 0.1. The decimal 0.1 means one tenth, not one hundredth. The decimal for 1% is 0.01. Reading the place values aloud helps prevent the common mistake of writing 5% as 0.5.

To convert a decimal to a percentage, ask how many hundredths it represents. The decimal 0.43 represents forty-three hundredths and is therefore 43%. The decimal 0.7 represents seven tenths, which is seventy hundredths, so it is 70%.

For a decimal with more places, the same relationship still works. The decimal 0.375 equals 37.5 hundredths, so it is 37.5%. Multiplying the decimal by one hundred finds the percentage number. This is more reliable than memorising a direction in which to move a point without understanding why.

Choose a representation that fits the calculation. To find 75% of 48, using 3/4 gives 48 ÷ 4 × 3 = 36. To find 43% of 250, using 0.43 gives 0.43 × 250 = 107.5. Neither representation is more correct; one may simply make a particular calculation easier.

You can also use proportional reasoning. If eighty of every hundred equal parts are wanted, then four of every five equal parts are wanted. This makes it clear why multiplying by 80/100, 4/5 or 0.8 must produce the same answer.

Activity: How Close Can You Get?

An estimate is a reasoned approximation, not a random guess. If you need the percentage equivalent of 2/9, compare it with familiar shares. A quarter of nine is 2.25, so 2/9 is less than 25%. One fifth of nine is 1.8, so 2/9 is greater than 20%. Its percentage lies between 20% and 25%.

Try the same approach with 7/12. Half of twelve is six, so the percentage is above 50%. Three quarters of twelve is nine, so it is below 75%. Because seven is only one above six, an estimate around 58% is more sensible than an estimate near 74%.

With a partner, choose positive whole numbers a and b with a < b. Both estimate the percentage represented by a/b within a few seconds, then calculate to check. Play several rounds. Explain the benchmarks you used after each round; the reasoning is more useful than speed alone.

Now consider a mixture whose millet-to-water ratio is 2:7. This compares millet with water. It does not yet compare millet with the whole mixture. The whole contains 2 + 7 = 9 equal ratio parts, so the millet share is 2/9 and the water share is 7/9.

Using 2/7 as the millet fraction would answer a different question: millet relative to the water alone. The denominator for an ingredient’s percentage of the mixture must include every ingredient counted in the mixture.

Keep the measurement basis consistent. A ratio by mass produces percentages by mass, while a ratio by volume produces percentages by volume. In our model, the ingredient volumes are taken as additive; real ingredients do not always behave that way when mixed.

Rounding only at the end usually keeps a calculation more accurate. Replacing 2/9 by 0.2222 at the start slightly changes the multiplier. The difference may be small for one calculation, but repeated early rounding can accumulate.

Finally, a recipe can be enlarged while keeping its FDP forms unchanged. A fruit-to-yogurt ratio of 1:3 means fruit is 1/4 = 0.25 = 25% of the total, whether the total mass is 200 g or 2 kg. The amount changes with scale; the three descriptions of its proportion stay equivalent.