Mixed Number Word Problems: Multiply or Divide?

Many division word problems are about equal groups, which are groups that all have the same size. In an equal-groups story, the number of groups times the size of one group gives the total.

Here is a ribbon story. You have $6$ pieces of ribbon, and each piece is $1\frac{1}{4}$ feet long. The total length is $6 \times 1\frac{1}{4} = \frac{6}{1} \times \frac{5}{4} = \frac{30}{4} = \frac{15}{2} = 7\frac{1}{2}$ feet.

Division undoes multiplication. If you know the total and one of the other two parts, you divide to find the missing part. The number you divide by is the divisor, and it must not be zero. This gives two kinds of division question:

• How many groups? Divide the total by the size of one group. The quotient is a count of groups.
• How big is one group? Divide the total by the number of groups. The quotient is the size of one equal share.

A count and a size are different things, so name the right one. The table asks both questions about the ribbon.

Two reminders before you solve. In a division, the dividend is the number being divided.

Why the reciprocal works. The reciprocal of a fraction that is not zero swaps its numerator and denominator, so the two multiply to $1$: $\frac{5}{4} \times \frac{4}{5} = \frac{20}{20} = 1$. Dividing by $\frac{5}{4}$ asks, "What number times $\frac{5}{4}$ gives the dividend?" Multiplying the dividend by $\frac{4}{5}$ undoes the multiplying by $\frac{5}{4}$ and leaves that number. So dividing by a fraction that is not zero gives the same result as multiplying by its reciprocal.

How to check. When the divisor is not zero and the quotient is exact (not rounded), the quotient times the divisor equals the dividend. If the quotient is rounded, the quotient times the divisor is only close to the dividend. If you share whole objects that cannot be cut and some are left over as a remainder, the check changes: the divisor times the whole-number quotient, plus the remainder, equals the dividend. Every quotient in this lesson is exact, with nothing left over.

Not every fraction story is a division story. In both examples above you knew the total, so you divided. Sometimes the total is the missing part. Then you know the number of groups and the size of one group, and you multiply them.

The number of groups does not have to be a whole number. Suppose one batch of muffins uses $2\frac{1}{4}$ cups of flour, and you bake $1\frac{1}{3}$ batches. One group is one batch, and its size is $2\frac{1}{4}$ cups. You have $1\frac{1}{3}$ groups. The total flour is $1\frac{1}{3} \times 2\frac{1}{4} = \frac{4}{3} \times \frac{9}{4} = \frac{36}{12} = 3$ cups.

A fraction less than $1$ can be the number of groups too. Half a batch is $\frac{1}{2}$ of one group, so it uses $\frac{1}{2} \times 2\frac{1}{4} = \frac{1}{2} \times \frac{9}{4} = \frac{9}{8} = 1\frac{1}{8}$ cups.

Look at the sizes. Baking $1\frac{1}{3}$ batches uses more flour than one batch, and baking half a batch uses less.

After you choose an operation, predict the size of the answer as a check. Each prediction holds only under its conditions.

A factor is a number you multiply by. Start with a positive quantity:

• A factor greater than $1$ gives a result greater than the starting quantity.
• A factor of $1$ gives a result equal to the starting quantity.
• A factor between $0$ and $1$ gives a result less than the starting quantity.

Division follows the same pattern, because dividing by a number that is not zero gives the same result as multiplying by its reciprocal. A positive number less than $1$, such as $\frac{2}{3}$, has a reciprocal greater than $1$, here $\frac{3}{2}$ or $1\frac{1}{2}$. So dividing a positive quantity by a positive number less than $1$ gives a quotient greater than the starting quantity. A number greater than $1$ has a reciprocal between $0$ and $1$, so dividing a positive quantity by it gives a quotient that is less.

If the starting quantity is $0$, the answer stays $0$: zero times any factor is $0$, and zero divided by any number that is not zero is $0$.

Takeaway. In an equal-groups story, find the three parts: the number of groups, the size of one group, and the total.

• If the total is missing, multiply.
• If the number of groups or the size of one group is missing, divide. The divisor must not be zero.

Then say what your quotient means: a count of groups, or the size of one equal share. Check an exact quotient by multiplying it by the divisor to get the dividend back.

Use size to check, not to choose. For a positive quantity, a factor greater than $1$ gives more, a factor of $1$ gives the same, and a factor between $0$ and $1$ gives less. Dividing a positive quantity by a positive number less than $1$ gives more.