### Video Transcript

If we subtract three times a certain number from each of the two terms in the ratio 33 to 19, the ratio becomes three to 10. What is the number?

So we’re told we need to subtract three times a certain number from the terms in the ratio 33 to 19. We’re going to use an algebraic method here. So let’s begin by defining our number to be equal to 𝑥. Then, we know we’re going to be subtracting three times this number, where three times is simply three 𝑥. Subtracting this from each of our two terms and the ratio becomes 33 minus three 𝑥 to 19 minus three 𝑥. But we’re told that when this happens, the ratio becomes three to 10. So 33 minus three 𝑥 to 19 minus three 𝑥 must be equal to three to 10. And this is really useful because whilst we have ratio symbols in here at the moment, we can form and solve an equation in 𝑥 by dividing one side of the ratio by the other.

In other words, if we divide 33 minus three 𝑥 by 19 minus three 𝑥, this is equivalent to dividing three by 10. So we form an equation. It’s 33 minus three 𝑥 over 19 minus three 𝑥 equals three over 10. And now we notice that these fractions are making our life a little bit difficult. So we’re going to multiply both sides of the equation by 19 minus three 𝑥 and by 10. When we do, our left-hand side becomes 10 times 33 minus three 𝑥. And our right-hand side is three times 19 minus three 𝑥. We’re now going to distribute our parentheses. Multiplying 10 by each term in the expression 33 minus three 𝑥 gives us 330 minus 30𝑥. Then, multiplying three by each term in the expression 19 minus three 𝑥 and we get 57 minus nine 𝑥.

To solve for 𝑥, let’s add 30𝑥 to both sides. When we do, our equation becomes 330 equals 57 plus 21𝑥. To isolate the term containing the 𝑥-variable, we’ll subtract 57, and that gives us 273 equals 21𝑥. Finally, since 21 is multiplying the 𝑥, we need to divide both sides of our equation by 21. So 𝑥 is 273 divided by 21, and this in fact is equal to 13. So if we subtract three times a certain number from each of the two terms in the ratio 33 to 19 and get a ratio of three to 10, that number must be 13.