Video: Comparing Values of Physical Quantities

In this lesson, we will learn how to compare values as ratios, fractions of each other, and as percentages of each other.

13:58

Video Transcript

In this video, we’re talking about comparing values of physical quantities. A physical quantity is anything that can be measured. And as we’ll see, there are several different ways of comparing these quantities. Three of the most common ways are using ratios, fractions, and percents. Let’s look at how these three mathematical tools let us compare values.

Now, for each of these three tools, we could see that there are two aspects to using them properly. On one hand, we’ll need to know how mathematically to calculate ratios, fractions, and percents. But there’s another aspect we’ll want to understand too. Oftentimes, say when we’re working an example exercise, the use of these tools is called for not in a mathematical equation, but in a sentence. For example, we might encounter a phrase like “find the ratio of 𝐴 to 𝐵,” where 𝐴 and 𝐵 are variables. Or similarly, we might see a phrase such as “what fraction of 𝑥 is 𝑦?”

When we encounter questions that are worded this way, we don’t just have to understand the mathematics of ratios, fractions, and percents, but we also need to understand how to translate from the sentences to the math. Let’s start out by considering how to do this using ratios. And then, we’ll move on to fractions and percents. If we want to find the ratio between two values and we can call those values 𝐴 and 𝐵, then there are a couple of different equivalent ways to write that mathematically. One way we could write it looks like this: 𝐴 colon 𝐵. This is one way we can use symbols to write the ratio of 𝐴 to 𝐵. But there’s another way to write this ratio that means the same thing mathematically. The ratio of 𝐴 to 𝐵 can also be expressed as 𝐴 divided by 𝐵. That is as a fraction.

So then, this expression, as well as this expression, both mean the ratio of 𝐴 to 𝐵. Each way of writing out the ratio has its own distinct advantage. When we write a ratio as a fraction like this, it makes it clear how to actually calculate this value. And then to see the advantage of writing a ratio using the colon notation, let’s imagine that our sentence changed a bit. Instead of finding the ratio of 𝐴 to 𝐵, what if we wanted to find the ratio of 𝐴 to 𝐵 to 𝐶. We now have three values to compare. In this case, using our colon notation, we could write that out like this. And note that we can’t write a ratio like this as a fraction.

So we see that mathematically ratios can be expressed in different ways. As far as translation goes, that is as far as taking a sentence and translating it to an equation, the important thing to remember is that whatever value is listed first — in this case, 𝐴 comes first — also comes first when we write our mathematical expression 𝐴 to 𝐵 or 𝐴 divided by 𝐵. So this is how we interpret ratios when we find them in problem statements. Now, let’s move on to talking about the second tool — fractions.

When we’re asked to solve for fractions, often it happens in a context like this. A sentence will read something like “what fraction of one variable — we’ve called it 𝑥 — is another variable — we’ve called it 𝑦?” Now, we already know in general what a fraction looks like. It’s one number — we can call it 𝑛 one — divided by another number — we can call it 𝑛 two. But the question is, when we’re asked something like what fraction of 𝑥 is 𝑦, where 𝑥 and 𝑦 are variables, which one we’ll need to know goes in the numerator of our fraction and which goes in the denominator. Our fraction in general will turn out differently, depending on which one we put where. So we’ll want to make sure that we know how to translate a sentence like this into a mathematical statement of a fraction.

Here’s one way to think about doing this. Whenever we’re asked a question like this, what we want to solve for is a fraction. And we could make up a symbol for that fraction. Let’s just call it capital 𝐹. Now, our sentence reads what fraction of 𝑥 one value is 𝑦 another value? We can translate this whole sentence into a mathematical equation that goes like this. 𝐹, the fraction we want to solve for, multiplied by our first variable, 𝑥, is equal to our second value — let’s call it 𝑦. In this equation, it’s the fraction 𝐹 we want to solve for. And we’re assuming that 𝑥 and 𝑦 are known values.

And if we know 𝑥 and 𝑦, then we see we can rearrange this equation to solve for 𝐹. 𝐹 is equal to 𝑦 divided by 𝑥. Notice by the way that this is a fraction. And depending on the values of 𝑦 and 𝑥, we might leave it written that way. Another option is to calculate 𝑦 divided by 𝑥 and write it as a decimal. Either way, we’re solving for the value of 𝐹. When it comes to fractions, this process of translating our sentence into an equation is an important one to remember. So let’s go over it once more.

Given a sentence like this in our question statement, we know that what we want to solve for is a fraction. And we’ve said that we can represent that fraction as a capital 𝐹. Of course, we don’t have to represent it that way, but that’s just one way we could do it. Now, our question says, what fraction of 𝑥 is 𝑦? In this case, this word “of” essentially translates into a multiplication symbol. And thinking about it, let’s actually make it a dot so we don’t confuse it with the variable 𝑥. So the phrase fraction of 𝑥 translates 𝐹 times 𝑥. And then, the word “is” could be translated as inequality, an equal sign. And then, on the other side of that equality is the symbol 𝑦. So a sentence like “what fraction of 𝑥 is 𝑦?” translates into the mathematical equation 𝐹, the fraction, times 𝑥 is equal to 𝑦.

And then, as we saw, we can rearrange this to solve for 𝐹 by dividing both sides of the equation by the value 𝑥. Just to show that the way we’re interpreting this sentence is correct, let’s plug in example values for 𝑥 and 𝑦. Let’s say that 𝑥 has a value of four and we’ll give 𝑦 a value of three. So our question is now reading, what fraction of four is three? When we do our translation from a sentence in words to an equation in math, we write this as 𝐹, the fraction we want to solve for, times four, that’s the value of 𝑥, is equal to three, the value of 𝑦. And then if we divide both sides of the equation by four, that term cancels out on the left-hand side. And what we find is that 𝐹, our fraction, is equal to three-fourths. We can leave this written as is, or we could express it as a decimal. Three-fourths is equal to 0.75. So our answer to the question “what fraction of four is three?” is three-fourths or 0.75.

When it comes to fractions, it’s this translation step we talked about, which is most helpful to remember: how to translate a phrase or sentence like what fraction of 𝑥 is 𝑦 into a mathematical equation.

Moving on to percents, we’ll find that these are similar to fractions. The main difference between the two comes from the meaning of this word “percent,” in other words, per 100. We may know that one cent is one one hundredth of a unit of money. And if we think about a mathematical equivalent to the word percent, we can think of it as dividing by 100. And we’ll see how this works more clearly in a moment. Now, say we’re working on an exercise and we encounter a phrase like this: What percent of 𝐴 is 𝐵? In order to show how calculating a percent is similar to calculating a fraction, let’s pretend just for the moment that this word is not percent, but instead is fraction.

So we’re imagining that the sentence actually reads “what fraction of 𝐴 is 𝐵?” Based on how we’ve learned to translate a sentence like this from words to mathematical symbols, we can write what fraction of 𝐴 is 𝐵 as capital 𝐹, the fraction we want to solve for, multiplied by 𝐴 is equal to 𝐵. And here, we’re assuming that 𝐴 and 𝐵 are known values. So this is the mathematical expression of this statement, if the statement used the word “fraction.” But now, let’s go back to the original wording “what percent of 𝐴 is 𝐵?” When we translate this sentence into a mathematical equation, in general, the form will stay the same. But the only thing that will be replaced is our capital 𝐹.

Now, let’s say that this percent that we’re trying to calculate is represented using a capital 𝑃. That’s the answer we want to solve for. Well, in that case, in place of capital 𝐹, which we could use to solve for a fraction, we’ll put capital 𝑃, the percent we want to solve for, divided by 100 percent. So if capital 𝑃 represents the percent we want to solve for, then the sentence “what percent of 𝐴 is 𝐵?” translates capital 𝑃 divided by 100 percent times 𝐴 is equal to 𝐵. And as a memory device, we can use this word “percent” both to remind us that 𝑃 needs to be divided by 100 and then also that it’s divided by 100 percent.

Once we’ve written our equation this way, we can isolate 𝑃, the percent we want to solve for, by multiplying both sides of the equation by 100 percent divided by 𝐴. Look at what happens when we do that. On the left-hand side, 100 percent cancels out, as does 𝐴, leaving us simply with 𝑃 on the left. And then on the right-hand side rearranging, we have 𝐵 divided by 𝐴 times 100 percent. Notice that the only difference between calculating a percent — what we’ve called capital 𝑃 — and a fraction — what we’ve called capital 𝐹 — comes down to multiplying by 100 percent here. If our sentence read what fraction of 𝐴 is 𝐵? then we could solve for that fraction using this equation. But if it reads what percent of 𝐴 is 𝐵? we can solve for it using this one. And the only difference between them, as we saw, is multiplying by 100 percent.

Now that we’ve seen how we interpret and also calculate ratios, fractions, and percents, let’s get a bit of practice with these ideas through an example.

The gravitational field strength at the surface of Earth is 9.8 newtons per kilogram. The gravitational field strength of the surface of the Moon is 1.6 newtons per kilogram. What is the ratio of the gravitational field strength at the surface of Earth to the gravitational field strength at the surface of the Moon? Answer to one decimal place.

Okay, so in this question, we’re comparing gravitational field strengths at two different locations: one is at the surface of Earth and the other is at the surface of the Moon. We’re given these strengths in units of Newtons per kilogram for both of these locations and we want to solve for their ratio. Specifically, we want to solve for the ratio of the gravitational field strength of the surface of Earth to the gravitational field strength of the surface of the Moon. In order to answer this question, we’ll need to interpret or translate this sentence into a mathematical expression.

To get started with doing that, let’s apply some symbols to the terms that we’ve been given. We’re told that the gravitational field strength at the surface of Earth is a certain value. Let’s call that field strength 𝑔 sub 𝐸. And then likewise, we’ll call the gravitational field strength at the surface of the Moon 𝑔 sub 𝑀. We’re told that 𝑔 sub 𝐸 is 9.8 newtons per kilogram and 𝑔 sub 𝑀 is 1.6 newtons per kilogram. We want to know the ratio of 𝑔 sub 𝐸 to 𝑔 sub 𝑀.

Now because of the way that these two terms — the gravitational field strength of the surface of Earth and the field strength of the surface of the Moon — are arranged in this sentence, we know that when we go to calculate this ratio, it will be 𝑔 sub 𝐸 divided by 𝑔 sub 𝑀. That’s because we’re finding the ratio of the field strength on Earth’s surface to that on the Moon’s surface. If the question had instead asked what is the ratio of the gravitational field strength at the surface of the Moon to the gravitational field strength at the surface of the Earth, then, in that case, we would need to reverse these two terms: 𝑔 sub 𝑀 would be on top and 𝑔 sub 𝐸 on bottom. But with our question written as it is, 𝑔 sub 𝐸 divided by 𝑔 sub 𝑀 is correct. That’s the ratio we want to solve for.

And to solve for that ratio, we’ll divide 9.8 newtons per kilogram by 1.6 newtons per kilogram. Notice by the way, that the units in these expressions cancel out, as they often do when we calculate a ratio. We’ll end up with a pure number. And it will be equal to 9.8 divided by 1.6. When we calculate that fraction, we find an exact answer of 6.125. But this isn’t our final answer because we want to give an answer to one decimal place. That means our answer will keep 6.1. And then we’ll round that one up if the next digit, the two, is greater than or equal to five. But since two is not greater than or equal to five, we’ll leave the one as it is. To one decimal place then, this is our answer. The ratio of the gravitational field strength at the surface of Earth to the gravitational field strength of the surface of the Moon is 6.1.

Let’s now summarize what we’ve learned about comparing values of physical quantities. Starting off, we saw that physical quantities are often compared using three tools: ratios, fractions, and percents. When considering ratios, we saw that a phrase like the ratio of 𝑥 to 𝑦 can be written mathematically in two equivalent ways. It could be written as 𝑥 colon 𝑦 or 𝑥 divided by 𝑦 as a fraction. We then considered fractions. And we interpreted phrases like “what fraction of 𝑥 is 𝑦?” When we let a capital 𝐹 represent the fraction we want to solve for, we saw that this statement is mathematically equivalent to 𝐹 times 𝑥 is equal to 𝑦. So we could find the fraction we wanted to solve for by dividing 𝑦 by 𝑥.

And lastly, considering percents, we saw that phrases like what percent of 𝑥 is 𝑦 can be interpreted mathematically this way. If we let capital 𝑃 be the percent we want to solve for, then the statement “what percent of 𝑥 is 𝑦?” translates to 𝑃 divided by 100 percent multiplied by 𝑥 is equal to 𝑦 so that the percent 𝑃 is equal to 𝑦 divided by 𝑥 times 100 percent. This is how we can interpret statements involving making comparisons using ratios, fractions, and percents.

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