**Division models: two ways of thinking of division**

Once you know your basic operations (addition, subtraction, multiplication, division), you will encounter story problems, also known as word problems, which require you to read a problem and decide which operation to perform in order to get the answer. There are key words here that often indicate which operation you will use. We will give you a list of them, but remember that for many problems, there may not be a key word, and you’ll have to use your best judgment in order to figure out what to do!

Here are the key words:

**For addition:**

In addition to, sum, total, more than, altogether, in all, combined, extra, raise, plus, both, additional

**For subtraction:**

How many more, difference, how many less, fewer, left (sometimes, left over), change, lost, decreased (by), less, remain, take away

**For multiplication:**

How many times, times, multiplied by, of, every, product, by, twice as much, three times as much (and so on), rate, at this rate, doubled, tripled (and so on), in all

**For division:**

How much/many will each receive?, divided among, split up between, per, ratio, percent, each, divide (or split) evenly, cut, average, share, quotient, equally (split, divided, etc)

**For telling that something equals another amount:**

is/are, yields

**In order to solve a story (word) problem successfully:**

- Read the
**entire problem**thoroughly - Make a list of the numerical (number) information you’ll need. If the numbers have units attached (for example, 12 inches), make sure you attach units in your list so you don’t get confused.
- Write out the number equation you’ll need to solve.
- Complete the solving process carefully.
- When you get your answer, reread the problem and ask yourself, “Does this answer make sense?”
- Remember to label your answer with the correct units, if needed.

## Example Story Problems

In this section, we’ll give you several examples of story (word) problems, starting with simple problems and working up towards more complex problems.

Nick had 8 toy trucks in his toy box. His friend Nathan brought over 3 more toy trucks. How many toy trucks did the boys have altogether?

What is the key word in this problem?

If you look back at the list of key words at the top of the page, you'll find that altogether listed as a key word.

altogether

Altogether is our key word. Now, what operation will we have to perform to get the answer to this problem?

**A**.

We know we’ll have to do addition, because altogether is a key word that means adding.

Now, what will our problem look like?

**C**.

We know we'll be adding together 8 + 3, because those were the two numbers mentioned in the problem.

What is 8 + 3?

**D**.

Therefore, our answer is 11 toy trucks altogether.

Now, let's try another one.

John had 15 books on his bookshelf. John’s dog, Buster, came in and slobbered all over four of them. How many books did John have left that were not slobbered on?

What is the key word in this problem?

If you look back at the list of key words at the top of the page, you'll find that left is listed as a key word.

left

Left is our key word. Now, what operation will we have to perform to get the answer to this problem?

**B**.

We know we’ll have to do subtraction, because left is a key word that means subtract.

This problem is a subtraction problem. Now, let’s get it set up. How will this problem look?

**D**.

We know we'll be subtracting 15 – 4, because those were the two numbers mentioned in the problem.

Now, perform the subtraction. What is 15 – 4?

**C**.

Our final answer is 11 books.

Now, let’s try a couple harder problems.

Dan is getting ready to go to a concert. He wants to figure out how many people will be there. He knows that there are 250 rows of seats, and each row has 40 seats in it. How many seats are there in the concert hall in all?

What is the key word in this problem?

If you look back at the list of key words at the top of the page, you'll find that in all is listed as a key word.

in all

In all is our key word. Now, what operation will we have to perform to get the answer to this problem?

**C**.

We choose multiplication because we see the keyword in all, but also because it makes sense. Essentially this may be seen as an addition problem, which is why the keyword is also in the addition section, but since the adding of the rows would all be the same, we can multiply to make the process faster.

This problem is a multiplication problem. Now, let’s get it set up. How will this problem look?

**A**.

We would use 250 x 40 because we decided that this is a multiplication problem. Since we want to figure out a total number of seats in the hall, we’re going to multiply the two given numbers together, as if we were calculating area.

Now, perform the multiplication. What is 250 x 40?

**C**.

Our final answer is that there are 10,000 seats at the concert Dan is attending.

Let’s look at one more example. Three friends go out to dinner. Near the end, they get the bill and they owe the restaurant .89. They want to split the bill evenly between the three of them. How much will each person pay?

What is the key word in this problem?

If you look back at the list of key words at the top of the page, you'll find that split evenly is listed as a key word.

{split|split the bill evenly|evenly|even}

Evenly is our key word. Now, what operation will we have to perform to get the answer to this problem?

**D**.

We choose division because we see the keywords “split” and “evenly.” Also, division makes sense because they want to divide the bill between three people. Because they’re splitting it up, we would choose division.

This problem is a division problem. Now, let’s get it set up. What would our equation be?

**A**.

We choose .89 / 3 because we know we have to split up the amount of money, .89, between the three friends, so we know we have to divide by three.

Now, perform the division. What is .89 / 3? (Round to the nearest cent)

**C**.

When we divide, we get an answer of 9.2966666, with a repeating 6 at the end. We want to round it to the nearest cent, which is the hundredths place after the decimal. We see that that number is already a 9, and a 6 after it means round up. However, we can’t make one place value a ten, so we increase the tenths digit by one, turning the 2 into a 3. If this doesn’t make sense, please read rounding numbers. Thus, your final answer is .30 after rounding.

Now, let’s go through some harder story (word) problems. All of the story problems we’ve done have had only one step, and we’ve been able to easily decide if they are addition, subtraction, multiplication, or division. However, some story problems have more than one step, involving more than one key word and/or operation. We’ll show you a few of these now.

Carly is making a dress. She needs 1 yard of yellow fabric, 1.5 yards of purple fabric, and .5 yards of green fabric. Yellow fabric costs .95 per yard, purple fabric costs .95 per yard, and green fabric is on sale for .00 per yard. How much will she spend in all if she buys just enough fabric to make her dress? (Ignore tax in your calculations). Click Next Step for the first part of the solution.

First, we have to figure out how much Carly is spending on each amount of fabric; then, we can use the key word “in all” which tells us that we need to add the amounts together for a final total. In order to figure out how much each piece of fabric costs, we need to multiply the price by the amount she needs to get a total.

First, let’s calculate the yellow fabric cost. She needs one yard, and it costs .95 per yard, so she’ll be spending .95.

Now, let’s calculate the purple fabric cost. She needs 1.5 yards, and it costs .95 per yard; therefore we have to multiply 1.5 times .95, which comes out to be .93 (we round to the nearest cent).

Finally, let’s calculate the green fabric cost. The green fabric is on sale for .00 per yard, and she needs .5 yards of it, so we multiply .00 times .5 and get .50.

Now, we have three money amounts (one for each color fabric) that we can now add together to get a total amount that Carly will spend. We know that we have to add these amounts together, like this:

.95 + .93 + .50 = .38

Thus, our final answer is that Carly will spend .38 on fabric for her dress.

Now, we’ll give you one to practice on. John is planning to carpet three rooms in his house. One room is 15 by 12 feet, one room is 17 by 14 feet, and the last room is 10 by 12 feet. John has 130 square feet of carpeting already. How much more carpeting does he need in order to carpet all three rooms?

First, you have to figure out how many square feet he has to carpet overall. That means we need to figure out the area of each room, and add those together. We multiply the dimensions together as follows:

15 x 12 = 180 ft2

17 x 14 = 238 ft2

10 x 12 = 120 ft2

Now, we have the area of each floor he has to carpet, so we can add these all together to find out the total amount of carpeting he needs.

180 + 238 + 120 = 538 ft2. This is the total amount John will need. However, the problem said that he already has 130 ft2 of carpet, so we need to figure out how much more he needs. Therefore, we need to subtract 130 from 538, and we get 408 ft2 leftover. This is how much more carpeting John will need to finish off his three rooms.

Final answer: 408 ft2.

{408 square feet|408|408 sq ft|408 ft2}

The Smith family is going to take a vacation to Florida. They live in Illinois, and have figured out that the trip is 1,150 miles from their house to the hotel in Florida. They get 28 miles per gallon of gas, and plan on travelling at an average rate of 60 miles per hour. Gas costs about .89 per gallon.

a) How long will it take them to get to Florida? (in hours)

For this part, you divide the total miles (1,150) by the speed they’re travelling (60 mph) and you would get 19.16666 (repeating). You would round the answer to 19.2 hours.

Final answer: 19.2 hours.

{19.2 hours|19.2 hrs|19.2 h}

b) How much money should they leave for gasoline (going one way)?

First, you would divide the total number of miles (1,150) by the amount of miles they get per gallon of gas (28); this gives you 41 gallons—the total amount needed for the trip. Then, you would multiply the number of gallons (41) by the cost per gallon of gas (.89) and round to the nearest cent, which gives you 8.49. This is the amount they should save for gas going one way.

Final answer: 8.49

{8.49}

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