The diagram shows the different stages to find the number 𝑏 that two-thirds is four-fifths of. Express 𝑎 in terms of two-thirds. What operation on number 𝑎 gives number 𝑏? And finally, find the value of 𝑏.
We’ll just take this one step at a time. First, we’re asked to express 𝑎 in terms of two-thirds. To do that, we need to compare the first and second image. In the first image, we have a block that’s equal to two-thirds. The second image has taken the block of two-thirds and divided it into four separate parts. 𝑎 is one of those four parts. So we can say that 𝑎 is equal to one-fourth of or one-fourth times the two-thirds we started with. 𝑎 equals one-fourth times two-thirds.
The next question says, what operation on a number 𝑎 gives number 𝑏? For this question, we will be looking at images two and three. 𝑎 was one of those four pieces. 𝑏 is five of those pieces, five of the 𝑎 values. And that means if we multiplied 𝑎 times five, we would find 𝑏. 𝑏 equals 𝑎 times five.
Our last question wants to know what the value of 𝑏 is. We know that 𝑏 equals 𝑎 times five. We need to substitute the value of 𝑎 with one-fourth times two-thirds. When we do that, we find out that one-fourth times two-thirds times five is the value of 𝑏. We can rewrite the whole number five as a fraction — five over one — and then we multiply.
When we’re multiplying our fractions, we multiply all the numerators and then multiply all the denominators. One times two times five equals 10. Four times three times one equals 12. This fraction can be reduced because the numerator and the denominator are divisible by two. 10 divided by two equals five. 12 divided by two equals six. What we found is that 𝑏 is equal to five-sixths.
Going back to our original statement, we can say two-thirds is four-fifths of five-sixths.