Video: Comparing Two-Step Expressions without Evaluating Them

Without doing any calculations, decide which expression has the highest value. [A] 1/9 × (115 + 709) [B] 1/10 × (115 + 709) [C] 1/2 × (115 + 709) [D] 1/4 × (115 + 709) [E] 1/5 × (115 + 709)

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Video Transcript

Without doing any calculations, decide which expression has the highest value. One-ninth multiplied by 115 plus 709. One-tenth multiplied by 115 plus 709. One-half multiplied by 115 plus 709. One-quarter multiplied by 115 plus 709. Or one-fifth multiplied by 115 plus 709.

The expressions that we’re given in this problem are all two-step expressions. They all contain a multiplication. And they all contain an addition. And we’re asked to find which one of these two-step expressions has the highest value. But we’re given a condition. We need to find the answer without doing any calculations. So, in other words, we need to be able to explain which has the highest value without actually working it out. We need to use reasoning and number sense.

In order to do this, we need to look carefully at these expressions and look at what’s the same and what’s different. We’ve already noticed that they’re all two-step expressions. But what else can we see? Well, perhaps one of the most obvious things to notice, and we would’ve heard this as we listened to the question being read aloud, is that each of the additions is the same, 115 plus 709. So, in fact, the only thing that’s different about our expressions is the fraction that they begin with.

So, as we decide which expression has the highest value, we’re going to have to look carefully at these fractions and think about how they behave when they’re multiplied. But before we do that, let’s carry on looking at these additions. What else do we notice about them? Well, each one is written inside parentheses, or brackets. What does this tell us? When we draw parentheses around an expression, it means that we need to do this part of the calculation first.

So, if we were working out the answer, we’d need to add 115 and 709 each time first, and then we just have a multiplication to do. We’d need to multiply the fraction by the total of what’s inside our parentheses. And understanding this can help us to solve the problem. The total of the values inside the parentheses is the same each time, but the fractions we begin with are all different. And so, the expression that has the highest value is going to be the one that begins with the fraction that has the highest value.

Although it looked more complicated, this question is just all about comparing fractions. So, let’s ignore the additions and just concentrate on the fractions for a moment. We have one-ninth, one-tenth, one-half, one-quarter, and one-fifth. Which is the largest? The first thing that we can see is that all the numerators in our fractions are the same. They’re all one. We call these fractions unit fractions. The only thing that’s different about our fractions, then, is the denominator. We just need to compare these now.

Now, a quick answer might be to look at all the denominators and to see which one’s the largest. 10 is the largest, and so surely one-tenth must be the largest fraction. We need to be careful about answering like this. And we need to give it a little more thought. The denominator in a fraction tells us how many equal parts a whole is split into. So, a large denominator like 10 tells us that the whole has been split into 10 equal parts. If we compare this with a smaller denominator like four, we can see that the whole has been split into less parts, so each part is larger.

In fact, there’s a rule here. Where all the numerators are the same, the larger the denominator, the smaller the fraction. All our numerators are the same. So, in order for us to find the largest fraction, we need to find the smallest denominator. And the smallest denominator is two. This means that the whole has only been split into two equal parts. This fraction is the largest.

And so, without doing any calculations, we’ve decided which expression has the highest value. Because the addition part of each expression is exactly the same, we simply had to compare our fractions to find which was the largest. And so, the expression with the highest value is one-half multiplied by 115 plus 709.

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