The ratio three-quarters to two-thirds is the same as the ratio one to 𝑘. Circle the value of 𝑘. The options are eight-ninths, nine-eighths, four-thirds, or five-sevenths.
These two ratios are equal to one another, which means they can each be scaled up or down to give the other ratio. Let’s consider the steps that we would need to go to in order to scale three-quarters up to a value of one.
If we first multiply three-quarters by four, then this would give us a value of three. We can think of this as just eliminating or cancelling out the denominator of four or we can think about the formal process of multiplying our fraction by an integer. Three-quarters multiplied by four can be thought of as three-quarters multiplied by the fraction four over one. The four in the numerator cancels with the four in the denominator, leaving three multiplied by one over one multiplied by one.
Multiplying the numerators of these fractions gives three and multiplying the denominators gives one. So we have three over one which is equal to three. So we’ve scaled three-quarters up to three by multiplying by four. But we remember our aim was to scale three-quarters to one. So we now need to divide three by three as three divided by three gives one.
Now whatever we do to one side of a ratio, we must also do to the other side in order to keep the ratio equivalent. So we need to take that value of two-thirds and first multiply it by four and then divide it by three to work out what the right-hand side of the ratio would be equal to.
To multiply two-thirds by four, we can again write four as four over one. We then multiply the numerators of these two fractions together giving eight and then multiply the denominators together giving three. So in the first stage of our scaling-up, we have the ratio three to eight over three.
Next, we need to divide eight over three by three. And to do this, we can think of the integer three as the fraction of three over one. We then recall that to divide by a fraction, we flip or invert that fraction and we multiply. So eight over three divided by three over one is equal to eight over three multiplied by one over three. We then multiply the numerators together giving eight and multiply the denominators together giving nine.
So two-thirds multiplied by four gives eight over three and eight over three divided by three gives eight over nine. We’ve shown then that the ratio three-quarters to two-thirds is equivalent to the ratio one to eight-ninths.
The value of 𝑘 then is eight-ninths.