We've seen how to express additive and multiplicative equations as comparison statements. In this lesson, we'll learn how to turn comparison statements into equations.
Let's write the following statement as an equation:
is larger than
Note the following:
The word "times" is not present. Therefore, this is an additive comparison.
The statement tells us that we must add an additional to to make This gives the following equation:
Finally, since we can swap the order of addition, we can also interpret our comparison statement as follows:
Some comparison statements involve subtraction. Let's see an example.
Which of the following equations shows that is smaller than
Let's examine our statement, highlighting some important words:
Which of the following equations shows that is smaller than
So, the statement compares the size of the number to the difference of and It can be interpreted as a comparison in the following way:
Which of the following equations shows that $16$ is $4$ larger than $12?$
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a
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$4\times 12 = 16$ |
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b
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$16\div 4 = 12$ |
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c
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$12-4 = 16$ |
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d
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$12+4 = 16$ |
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e
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$12\times 4 = 16$ |
Which of the following equations shows that $25$ is $5$ smaller than $30?$
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a
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$25\div 5 = 30$ |
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b
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$30+5 = 25$ |
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c
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$5\times 25 = 30$ |
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d
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$30-5 = 25$ |
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e
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$25\times 5 = 30$ |
Let's express the following statement as an equation:
is times larger than
Note the following:
The word "times" is present. Therefore, this is a multiplicative comparison.
The statement tells us that we must add copies of to make This gives Since multiplication is repeated addition, we can write this equation as follows:
Finally, since we can swap the order of multiplication, we can also interpret our comparison statement as follows:
Every multiplicative comparison can be expressed as two separate equations.
Which of the following equations shows that is times larger than
Let's examine our statement, highlighting some important words:
Which of the following equations shows that is times larger than
So, the statement compares the size of a product to its factors and It can be interpreted as a comparison in the following two ways:
From the given options, the correct answer is
Which of the following equations shows that $48$ is $12$ times larger than $4?$
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a
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$12 \times 4 = 36$ |
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b
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$4 \times 12 = 12$ |
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c
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$4 + 12 = 48$ |
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d
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$48 \times 12 = 4$ |
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e
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$4 \times 12 = 48$ |
Which of the following equations shows that $24$ is $6$ times larger than $4?$
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a
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$4 \times 6 = 20$ |
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b
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$6 \div 4 = 24$ |
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c
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$6\times 4 = 24$ |
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d
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$6 + 4 = 24$ |
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e
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$24 \times 4 = 6$ |
It's common in math to use letters to represent unknown numbers.
For example, suppose we want to answer the following question:
Which number is times larger than
This is a multiplicative comparison statement. We can express this statement using an equation as follows:
Here, we use the letter to represent the unknown product.
To find the value of we simply compute This gives
We usually put the letter on the left-hand side:
Therefore, we conclude that the number is times larger than
What number is times larger than
Let's examine our question statement, highlighting some important words:
What number is times larger than
So, the statement compares the size of a product to its factors and Here, the letter represents the unknown product.
Therefore, our statement can be interpreted in the following two ways:
Calculating the products, we have
This means that
Which of the following equations gives the number that is $4$ times larger than $6?$
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a
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$4 \times 6 = 32$ |
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b
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$6 \times 4 = 32$ |
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c
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$6 \times 4 = 24$ |
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d
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$6 + 4 = 10$ |
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e
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$4 + 6 = 12$ |
What number is $5$ times larger than $6?$
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a
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$30$ |
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b
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$25$ |
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c
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$36$ |
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d
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$11$ |
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e
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$17$ |
The number is times larger than which number?
Let's examine our statement, highlighting some important words:
The number is times larger than which number?
So, the statement compares the size of a product to its factors, and Here, represents the unknown second factor.
Therefore, our statement can be interpreted in the following two ways:
Therefore, and we have
So, the correct answer is
Which of the following equations shows that $42$ is $7$ times larger than some other whole number?
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a
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$8 \times 7 = 42$ |
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b
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$6 \times 7 = 42$ |
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c
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$8 + 7 = 42$ |
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d
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$7 \times 7 = 42$ |
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e
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$35 + 7 = 42$ |
The number $28$ is $4$ times larger than which number?
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a
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$24$ |
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b
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$14$ |
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c
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$7$ |
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d
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$12$ |
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e
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$9$ |