Have you ever had to figure out a dilemma involving money? Take a look at this situation.

The City Orchestra received a total of $1,891.50 in donations. This needs to be divided evenly among six different departments. How much will each department receive?

Use the information in this Concept to help you solve this problem using decimals and estimation.

### Guidance

To estimate products and quotients with decimals, you need to first round the numbers so that they are easier to work with. To round to the nearest whole number, look at the digit in the tenths place. If it is less than 5, round down. If it is 5 or greater, round up.

**Remember that an** *estimate***is an answer that is not exact, but is approximate and reasonable.**

Take a look at this one.

Estimate the product: \begin{align*}11.256 \times 6.81\end{align*}

First, we round the first number. Since there is a 2 in the tenths place, 11.256 rounds down to 11.

Now round the second number. Since there is an 8 in the tenths place, 6.81 rounds up to 7.

Now multiply the rounded numbers.

\begin{align*}11 \times 7 = 77\end{align*}

**A good estimate for the product is 77.**

Here is another one.

Estimate the quotient: \begin{align*}91.93 \div 4.39\end{align*}

First we round the first number. Since there is a 9 in the tenths place, 91.93 rounds up to 92.

Now round the second number. Since there is a 3 in the tenths place, 4.39 rounds down to 4.

Now divide the rounded numbers.

\begin{align*}92 \div 4=23\end{align*}

**A good estimate for the quotient is 23.**

**Did you notice which numbers we multiplied? We multiplied the whole digits or the digits that were leading the entire number. We did the same thing when we divided.**

Yes. We work with the digits that are in the “lead”. With those digits, we can find a reasonable estimate.

Estimate by using leading digits

#### Example A

\begin{align*}4.237 \times 12.123\end{align*}

**Solution: \begin{align*}48\end{align*} 48**

#### Example B

\begin{align*}16.123 \div 4.00012\end{align*}

**Solution: \begin{align*}4\end{align*} 4**

#### Example C

\begin{align*}162.003 \times 2.137\end{align*}

**Solution: \begin{align*}324\end{align*} 324**

Now let's go back to the dilemma from the beginning of the Concept.

**Notice that the key word “each” tells us that we are going to need to divide. The money is being split up and that means division.**

Divide to find the amount each department will receive: \begin{align*}1891.5 \div 6 = 315.25\end{align*}

**Each department will receive $315.25.**

### Guided Practice

Here is one for you to try on your own.

Estimate using leading digits.

\begin{align*}120.0045 \div 6.237\end{align*}

**Solution**

First, we take only the leading digits and rewrite this problem.

\begin{align*}120 \div 6\end{align*}

Now our work is quite simple.

\begin{align*}120 \div 6 = 2\end{align*}

**Our estimate is \begin{align*}2\end{align*} 2.**

### Video Review

### Explore More

Directions: Estimate each product or quotient by using leading digits.

1. \begin{align*}35.0012 \div 5.678\end{align*}

2. \begin{align*}5.123 \times 11.0023\end{align*}

3. \begin{align*}12.0034 \div 4.0012\end{align*}

4. \begin{align*}12.123 \times 3.0045\end{align*}

5. \begin{align*}48.0012 \div 12.098\end{align*}

6. \begin{align*}13.012 \times 3.456\end{align*}

7. \begin{align*}33.234 \div 11.125\end{align*}

8. \begin{align*}12.098 \times 2.987\end{align*}

9. \begin{align*}4.769 \times 8.997\end{align*}

10. \begin{align*}14.98 \div 7.002\end{align*}

11. \begin{align*}24.56087 \div 8.0012\end{align*}

12. \begin{align*}45.098 \div 5.0098\end{align*}

13. \begin{align*}9.0987 \times 9.0001\end{align*}

14. \begin{align*}34.021 \times 4.012\end{align*}

15. \begin{align*}21.0098 \times 2.0987\end{align*}

16. \begin{align*}14.231 \times 3.7601\end{align*}

17. \begin{align*}144.0056 \div 12.0112\end{align*}