Given that 𝑎 to 𝑏 equals three to two and 𝑎 minus 𝑏 equals 49, calculate the value of 𝑎.
We’re given information about the ratio of 𝑎 to 𝑏 alongside an equation that links 𝑎 minus 𝑏 with some constant. In order to use all of this to find the value of 𝑎, we will need to rewrite this equation so it’s purely in terms of 𝑎. So how do we do that?
Well, we’re going to use the fact that the ratio of 𝑎 to 𝑏 is equal to three to two. This will allow us to find a value of 𝑎 divided by 𝑏. If the ratio of 𝑎 to 𝑏 is three to two, then when we divide 𝑎 by 𝑏, we’re going to need to divide three by two. So 𝑎 divided by 𝑏 is equal to three over two.
Alternatively, we can find the reciprocal of both sides here. In other words, 𝑏 over 𝑎 is equal to two over three. Then, we’re going to make 𝑏 the subject because this will allow us to find an expression for 𝑏 in terms of 𝑎 that we can substitute into the equation. To do so, we multiply through by 𝑎. That gives us 𝑏 equals two-thirds 𝑎.
We now replace 𝑏 in the equation 𝑎 minus 𝑏 equals 49 with two-thirds 𝑎. So 𝑎 minus two-thirds 𝑎 is equal to 49. Well, one 𝑎 is equivalent to three-thirds 𝑎. So we have three-thirds 𝑎 minus two-thirds 𝑎, which is one-third 𝑎. So one-third 𝑎 is equal to 49. To make 𝑎 the subject, to solve for 𝑎, we divide through by one-third. And that’s equivalent to multiplying by three. 49 times three is 147. So we found our value of 𝑎. 𝑎 equals 147.