Understanding numbers

You need to decide which of these numbers is the smallest, which is the next smallest, which is the next smallest, and which is the largest. This is difficult, because all the numerators are different, which makes it difficult to compare the fractions. If you express the fractions as equivalent fractions with the same denominator, you can then compare the numerators to see which are the smallest, and so on.

The best numerator to choose is the lowest common multiple of the numerators. In this case, the lowest common multiple of 4, 6, 7, and 8 is 168. This means that 168 is the smallest number which 4, 6, 7, and 8 all divided into, giving a whole number result.

To convert 7/8 to an equivalent fraction with a denominator of 168, you multiply the numerator and denominator by 21. This gives 147 over 168. To convert 6/7 to an equivalent fraction with a denominator of 168, you multiply the numerator and denominator by 24. This means that 6/7 is the same as 144 over 168. To convert 3/4 to an equivalent fraction with a denominator of 168, you multiply numerator and denominator by 42. This means that 3/4 is 126 divided by 168. To convert 5/6 to an equivalent fraction with a denominator of 168, you multiply the numerator and denominator by 28. This means that 5/6 is 140 divided by 168.

So comparing the numerators, 3/4 is fourth, 5/6 is third, 6/7 is second, and 7/8 is the largest.

You can always check your answers with a calculator if you have one. Here, all answers are given to three decimal places. 7/8 is 0.875. 6/7 is 0.857. 3/4 is 0.750. And 5/6 is 0.833. So 3/4 is fourth, 5/6 is third, 6/7 is second, 7/8 is largest.

To add these four fractions together, you need to express them all as having the same denominator. The denominator of a fraction tells you what type of fraction you have, and you can only add together things that are of the same type. Here, we have 8ths, 7ths, 4ths– or quarters– and 6ths. They’re all different. And so adding them together doesn’t make any sense. However, if we will express them with the same denominator using the method from before, 8/7 is 147 over 168, 6/7 is 144 over 168, 3/4 is is 126 over 168, and 5/6 is 140 over 168. As these equivalent fractions all have the same denominator, you can add them together.

This equals 147 plus 144 plus 126 plus 140, all divided by the common denominator of 168. This equals 557 over 168.