Introduction to Division:

Sometimes we have a direct translation from English to math, since the symbol ÷ abbreviates the phrase "divided by.'' None of that shady "together with" or "product of" business.

Like so: Five divided by three means 5 ÷ 3.

Here are some phrases that aren't quite as direct, but still straightforward. In other words, they may not make eye contact with you, but you still believe what they're saying.

With these phrases, the first number mentioned goes on top of the line, while the second number mentioned goes below the line. (The second number would totally win in a limbo competition.)

  • The ratio of six and seven is .
     
  • The quotient of seventy and thirteen is .

We can think of the word of as meaning either multiplication or division. How's that for confusing? Or, put another way, how's of that of for of confusing?

Sample Problem

What is one-third of seven?

If we translate of as multiplication, we get .

If we translate of as division, we get .

Thankfully,  and  mean the same thing. As the saying goes,  of one,  of another. Although these two translations look different, they give us equivalent expressions.

When translating the word of, look at the numbers and other words involved to decide if it's more appropriate to translate as multiplication or division. Yes, you'll need to use some logical reasoning here; it won't always be spelled out for you. When words like one-half, one-third, or one-fourth are floating around next to the of, you can think of this as multiplication by a fraction, or you can choose to translate it as a division problem. We'll even spell it out for you: A D-I-V-I-S-I-O-N P-R-O-B-L-E-M.

Division Practice:

Example 1

One-half of nine is what?


Example 2

One-sixth of twenty is what?


Exercise 1

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

The product of four and five.


Exercise 2

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

Eight times y.


Exercise 3

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

Forty-two divided by fourteen.


Exercise 4

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

Four-fifths of twelve.


Exercise 5

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

Seven doubled.


Exercise 6

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

Three by five.


Exercise 7

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

The ratio of six and fourteen.


Exercise 8

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

One-half of eleven.


Exercise 9

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

The quotient of eleven and one hundred.


Exercise 10

Translate the following English phrase into mathematical symbols, using multiplication or division as appropriate:

Thirty percent of 150.


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