Question

An infinite number of ratios are equivalent to 3:8.How can that be possible,when the multiplication table only goes to 9?

Answer 1

Explanation:

If you multiply both 3 and 8 by the same number, you get an equivalent rate to 3:8.

For example, multiply both 3 and 8 by 5.

3 * 5 = 15

8 * 5 = 40

The rate 15:40 is equivalent to 3:8.

Since there is an infinite number of different numbers that you can multiply both 3 and 8 by, there is an infinite number of equivalent rates to 3:8.

The multiplication tables do not go only to 9. We only normally memorize the multiplication tables to 9 or 10 because that allows you to do any multiplication on paper. If you want, you can write the multiplication tables to 100 or 1000 or any other number you choose. There is an infinite number of multiplication tables, but we only use very few of them.

Answer 2

Certainly, you know how to multiply by numbers larger than 9. If you're doing it by hand (not using a calculator), you make use of place value and partial products. The numbers in the multiplication table, together with appropriate procedures, let you multiply by any number you like.

If we want to multiply this ratio by 85, we have

... 3×85 : 8×85

... = 3×(8×10 +5) : 8×(8×10 + 5)

... = (3×8×10 +3×5) : (8×8×10 + 8×5)

... = (24×10 + 15) : (64×10 + 40)

... = (240 + 15) : (640 +40)

... = 255 : 680 . . . . . without using multiplcation table values beyond 9

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The factor of 85 that we arbitrariy chose could have been any of an infinite number of factors, including negative and fractional values.