Expanding and reducing fractions: Introduction

​​Is this a fraction or does this mean division? You could say that fractions are divisions that have not been calculated yet. When you solve simple problems, you can usually treat all fractions as divisions, plug them into a calculator, and get decimals for an answer. But some fractions cannot be represented as decimals without rounding, and when you get further on in mathematics, problems start to look like this, and representing the fractions as decimals will not be possible. There are a few simple tricks to use if you want to make hard fractions easy.

Here are two of them: expanding and simplifying. The first thing you need to know about fractions is this. Listen carefully: The value of a fraction doesn't depend on the actual numbers in its denominator and numerator, but on the relationship between them, how big they are in proportion to each other. Two thirds is the same as four sixths. That's easy to see, because 2 is the same proportion of 3, as 4 is of 6.

And as 8 is of 12, and as 3 million is of 4.5 million. ​ ​Fractions show ratios, relationships, proportions between numbers. Because a fraction's value, the quotient, is the same as long as the ratio between the numbers is the same, you can increase or decrease both the denominator and the numerator without changing the fraction's value. But you must do it by the same factor. Here we have doubled both the numerator and the denominator, and doubled again. And then multiplied both the numerator and the denominator by 375,000, and all four fractions still have the same value, the same quotient.

Plug them into a calculator if it's hard to believe. This is called expanding a fraction. As with most mathematical operations, you can do what we just did with multiplication backwards, using division. ​In this case, we simplify a fraction. The fraction twenty-one twenty-eighths can be simplified by dividing both the denominator and the numerator by the same number. Both twenty-one and twenty-eight are in the seven times table, so it will be neat and simple if we simplify by seven.

3 is the same proportion of 4 as 21 is of 28. The numerator and the denominator have the same relationship to each other. The ratio is the same. It's easy to simplify or expand by tens, hundreds, or thousands. You just move the decimal point to the right or to the left.

Take 250 thousandths. You can cross out a zero above and below. Now you've divided both the numerator and the denominator by ten to get 25 hundredths. If you want to go further and calculate the quotient, move the decimal point two places to the left. It's the same as dividing by hundred.

Zero point two five divided by one equals zero point two five. ​ ​You can multiply and divide by any number as long as you do the same thing for both the denominator and the numerator, but be careful. There is a trap. Here is twenty-five thirtieths. 25 divided by 30 is approximately 83 hundredths. Let's say you want to expand this fraction, to ninetieths.

How do you get 90 in the denominator? The trap is not to add or subtract. Look, if you add 60 to both the denominator and the numerator, you would get 90 in the denominator. But the ratio between the denominator and the numerator will not be the same. 85 over 90 is approximately 94 hundredths, not 83.

So, let's get rid of that. Expanding and simplifying only work with multiplication and division, for it is only then that you can keep the same ratio between the denominator and the numerator. Try this instead: by what number can you multiply or divide 30 to get 90? Three, of course. Thirty times 3 is 90.

Multiply both the the denominator and the numerator times 3. Then we have 90 in the denominator as we planned, and the ratio between the denominator and the numerator stays the same. A fraction's value depends on the ratio between the numerator and the denominator. If you multiply or divide both the denominator and the numerator by the same number, you keep the same ratio, and therefore, the same value. This is called expanding or simplifying fractions.