A percent is a fraction where the denominator, i.e. the whole, is 100. In the diagram, a 100-centimeter ruler has been used to represent the whole.
We have this diagram. And we have three different parts to this question. We’ll consider each one in turn.
Part one, what is the length of the shaded part in centimeters? What percent of the ruler is this?
To give us space to work out the first part, we’ll clear out the other two questions and bring them back when we’re ready. We want to know the length of the shaded part of this ruler measured in centimeters. The first thing we do is identify the shaded part. And we know that the entire ruler is 100 centimeters. The shaded part extends from zero to 30. Since the ruler’s being measured in centimeters, the shaded part is then 30 centimeters.
The second part asks us, what percent of the ruler is this? If we go back to our introduction, it reminds us that a percent is a fraction where the denominator is 100, where the whole is 100. If our shaded portion, our part, is 30 centimeters and the whole is 100 centimeters, we have a fraction where the denominator is 100. 30 centimeters out of 100 centimeters written as a percent would be 30 percent. And that means we’re now ready to consider part two of this question.
If now we use meter as a length unit and the ruler is one meter long, what is the length of the shaded part in meters?
What’s happening now is that the whole we’re switching from a unit of 100 centimeters to a unit of one meter. But our shaded portion is not changing. And there are a few ways we can think about this. Earlier, we had a fraction of the shaded part to the whole that was 30 centimeters over 100 centimeters. And now, we want to know the shaded amount in meters over one. Since the whole unit is one, we can know that the shaded portion is going to be less than one.
To go from 100 to one, to convert centimeters to meters, we divide by 100. And just as any time we work with fractions, if we divide by 100 in the denominator, we need to divide by 100 in the numerator, which means the numerator should be 30 divided by 100, which is 0.30. Here, we’re not looking for a ratio. We’re just looking for a length in meters. And so, we can say that the length in meters of the shaded portion is 0.30 meters. However, a simplified form would be instead of saying thirty hundredths, we could say three-tenths. 30 centimeters in length is equal to three-tenths of a meter.
And this makes sense to us because if we divided the ruler into 10 equal segments each segment made up of 10 centimeters, the shaded portion takes up three of those tenths. And now, we’re ready to consider the last part of this question.
The ratio of the length of the shaded part to that of the whole ruler can be expressed as a fraction, a percent — which is a fraction where the denominator is 100 — or as a decimal. In the latter case, the decimal can be interpreted as the length of the shaded part when the whole is one. Give this ratio as a decimal.
We want the ratio of the length of the shaded part to that of the whole ruler. We know the whole ruler measures one meter and that in meters, the shaded portion is 0.3. We’ve also seen this written as the fraction 30 centimeters over 100 centimeters. But since we’re looking for the case when the whole is one, we’ll use this ratio 0.3 meters to one meter.
But this 0.3 meters over one meter is not a ratio because ratios do not have units. We can take this fraction of 0.3 meters to one meter and convert it to a ratio by simply saying that the ratio of the shaded to the whole is 0.3 to one. And we can simplify that to say 0.3. This first version would be the ratio given as a fraction, but we wanted the ratio given as a decimal: 0.3.