Video Transcript
When the multiplicative inverse of a number is added to the consequent of the ratio six to five, the result is the ratio eight to seven. What is the number?
Let’s first remind ourselves what we mean by the multiplicative inverse of a number and the consequent of a ratio. Let’s begin with the multiplicative inverse. Say we have a number 𝑥. When we multiply that number by its reciprocal one over 𝑥, we get one. 𝑥 and one over 𝑥 are multiplicative inverses of one another. When we multiply them together, we get the value of one. Now, let’s think about the consequent. Say we have a ratio 𝑎 to 𝑏. The second number in this ratio 𝑏 is the consequent. As a side note, the first number 𝑎 is called the antecedent. So what does this mean for our ratio?
Let’s define the multiplicative inverse of our number to be equal to 𝑦. Then the consequent is the number five. So adding the multiplicative inverse of our number to the consequent of our ratio and the ratio becomes six to five plus 𝑦. But then, we’re told the result is the ratio eight to seven. So six to five plus 𝑦 is equal to eight to seven. And this is really useful because we can now divide one part of our ratio by the other and form and solve an equation for 𝑦.
Let’s divide the consequent of each part of the ratio by the antecedent. When we do, we find that five plus 𝑦 over six equals seven-eighths. To make 𝑦 the subject, let’s multiply through by six. That gives us five plus 𝑦 equals seven-eighths times six. Then, we cross cancel by a factor of two. And the right-hand side is seven-quarters times three or 21 over four.
Next, we’re going to subtract five from both sides. So 𝑦 is 21 over four minus five. But five is equivalent to 20 over four, twenty-quarters. So 𝑦 is 21 over four minus 20 over four, which is simply one-quarter. So, 𝑦 is a quarter; that’s the multiplicative inverse of our number. But we’re trying to find the original number. So we want the reciprocal of 𝑦; we want one over 𝑦, which is one over one-quarter or simply four.
So given the information in this question, we know that the number they’re interested in is four.
