Multiplication and fraction conversions

Multiplication with multiple digits

When you multiply multi-digit numbers by hand, it helps to break the problem into smaller steps and work with one digit at a time. Lining the numbers up by place value keeps everything organized so you can combine the partial products at the end. Let’s look at this in action.

Example:

What is $12\times 237?$

First, line up the numbers by place value.

$$\begin{array}{r}237\\ \times 12\end{array}$$

Now multiply $237$ by the ones digit of $12$, which is $2$.

• $2\times 7=14$.
• Write $4$ in the ones place and carry the $1$.

$$\begin{array}{r}1\\ 237\\ \times 12\\ 4\end{array}$$

Next, multiply the tens digit and include the carried value.

• $2\times 3=6$.
• Add the carried $1$: $6+1=7$.
• Write $7$ in the tens place.

$$\begin{array}{r}237\\ \times 12\\ 74\end{array}$$

Then multiply the hundreds digit.

• $2\times 2=4$.

$$\begin{array}{r}237\\ \times 12\\ 474\end{array}$$

Now multiply $237$ by the next digit of $12$, which is $1$.

Because this $1$ is in the tens place, the result shifts one place to the left. Start by writing a $0$ in the ones place.

(Note: each new digit on the bottom shifts the partial product one place to the left, so each new line starts with one more $0$ than the line above it.)

$$\begin{array}{r}237\\ \times 12\\ 474\\ 0\end{array}$$

Now multiply each digit by $1$.

• $1\times 7=7$
• $1\times 3=3$
• $1\times 2=2$

$$\begin{array}{r}237\\ \times 12\\ 474\\ 2370\end{array}$$

Finally, add the partial products.

$$\begin{array}{r}474\\ +2370\\ 2844\end{array}$$

Our final answer:

$$2844$$

Multiplying with decimals

Multiplying decimals works the same way as multiplying whole numbers. The only difference is where the decimal point goes in the final answer. Let’s walk through an example.

Example:

What is $3.25\times 5.6?$

First, ignore the decimals and multiply $325$ by $56$.

$$\begin{array}{r}325\\ \times 56\end{array}$$

Next, follow the same multi-digit multiplication steps: $325\times 6=1950$, and $325\times 5=1625$, shifted one place to the left to get $16250$.

$$\begin{array}{r}325\\ \times 56\\ 1950\\ 16250\end{array}$$

Now add the partial products.

$$\begin{array}{r}1950\\ +16250\\ 18200\end{array}$$

Now place the decimal point.

• $3.25$ has $2$ digits after the decimal.
• $5.6$ has $1$ digit after the decimal.
• Total: $2+1=3$ digits after the decimal in the product.

Move the decimal three places to the left (because there are three total decimal places in the original numbers):

• 18200 becomes 18.200

Our final answer is:

$$3.25\times 5.6=18.2$$

No calculator is allowed on this subtest, so count the decimal places directly from the full product, $18200$, before dropping any trailing zeros. Check the placement by estimating: $3\times 6=18$, so $18.2$ is reasonable, while $1.82$ or $182$ are not.

Improper and mixed numbers

Fractions represent parts of a whole, so the numerator is often smaller than the denominator. When the numerator is equal to or larger than the denominator, the fraction represents one whole or more. This is called an improper fraction.

Example 1:

$\frac{3}{4}$ is a proper fraction, while $\frac{3}{2}$ and $\frac{4}{4}$ are improper fractions.

You can also rewrite an improper fraction as a mixed number, which shows a whole number plus a fraction. To convert an improper fraction to a mixed number:

• Divide the numerator by the denominator
• The quotient becomes the whole number
• The remainder becomes the new numerator
• The denominator stays the same

Example 2:

$\frac{17}{5}$ becomes $3\frac{2}{5}$ because $5$ goes into $17$ three times ($5\times 3=15$) with a remainder of $2$.

If the remainder is $0$, the result is a whole number: $\frac{4}{4}=1$ and $\frac{12}{4}=3$.

Converting a mixed number into an improper fraction

To convert a mixed number into an improper fraction:

• Multiply the denominator by the whole number
• Add the numerator
• Place the result over the original denominator

Example:

$5\frac{3}{4}$ becomes $\frac{23}{4}$ because $4\times 5=20$, and $20+3=23$. So $23$ is the new numerator, and the denominator stays the same.

Check: $23÷4=5$ remainder $3$, which gives back $5\frac{3}{4}$.