When Dylan‘s baby sister was born last month she weighed 6 pounds 8.7 ounces, or 6.54375 pounds. Dylan‘s mom said that his sister will probably triple her weight by the time she is a year old. How could Dylan estimate what his sister will weigh when she is one year old?

In this concept, you will learn to estimate decimal products by multiplying leading digits.

### Estimating Decimal Products

Recall that when you **estimate** you are finding an approximate solution to a problem. One good time to use estimation is when you want to check your answer to a problem. If you estimate a solution to a problem either before or after you solve it, you can see whether or not your answer is realistic.

When you are working with decimal numbers with many digits, one way to estimate their product is to multiply only their two leading digits.

Here are the steps for multiplying decimal numbers using leading digits.

- Identify the two leading (left-most) digits of each number, preserving the location of the decimal point. Note that if 0 is the only digit to the left of a decimal point, it does not count as one of the leading digits.
- Multiply the leading digits using your steps for decimal multiplication.

Here is an example.

Estimate the product of

.First, identify the two leading digits of each number.

- The leading digits of 6.42 are 6.4
- The leading digits of 0.383 are 0.38

Note that because 0 is the only digit to the left of the decimal point in the second number, it does not count as one of the leading digits.

Now, multiply just as if you were multiplying whole numbers. Because the original numbers have 3 digits total after their decimal points, insert a decimal point into your answer so that it has 3 digits to its right.

\begin{align*}\begin{array}{rcl} & 6.4 \\ \times & \underline{.38} \\ & \ 512 \\ + & \underline{1920} \\ & 2.432 \end{array}\end{align*}The answer is the product of 6.42 and 0.383 is approximately 2.432.

Here is another example.

Find the product. Then estimate to confirm your solution.\begin{align*}22.17 \times 4.45\end{align*} .

First, notice that this problem asks you to do two things. You will need to multiply and then estimate the solution. Start by multiplying just as if you were multiplying whole numbers. Because the original numbers have 4 digits total after their decimal points, insert a decimal point into your answer so that it has 4 digits to its right.

\begin{align*}\begin{array}{rcl} & \ 22.17 \\ \times & \underline{\;\;\;\; 4.45} \\ & \ 11085 \\ & \ {88680} \\ + & \underline{886800} \\ & 98.6565 \end{array}\end{align*}Next, estimate the answer by multiplying only the leading digits. Identify the two leading digits of each number.

- The leading digits of 22.17 are 22
- The leading digits of 4.45 are 4.4

Now, multiply the leading digits.

\begin{align*}\begin{array}{rcl} & \ 22 \\ \times & \underline{\; 4.4} \\ & \ \ 88 \\ + & \underline{\; 880} \\ & 96.8 \end{array}\end{align*}Next, compare your answer with your estimate. Your answer of 98.6565 is close to your estimate of 96.8. This means you can be confident that your answer is correct.

The answer is that the exact product of \begin{align*}22.17 \times 4.45\end{align*} is 98.6565, while the estimate of the product is 96.8.

### Examples

#### Example 1

Earlier, you were given a problem about Dylan and his baby sister.

When she was born she weighed 6.54375 pounds and she will probably triple her weight by the time she is a year old. Dylan wants to estimate how much she will weigh when she is a year old.

First, Dylan needs to realize that if his sister triples her weight that means her original weight will be multiplied by 3. Dylan is trying to find the product of

.Now, Dylan can estimate the product by multiplying the leading digits. He should identify the two leading digits of each number.

- The leading digits of 6.54375 are 6.5
- The leading digits of 3 are 3

Note that because 3 only had one digit to start with, it only has one leading digit!

Next, Dylan should multiply just as if he was multiplying whole numbers. Because the original numbers have 1 digit total after their decimal points, he should insert a decimal point into his answer so that it has 1 digit to its right.

\begin{align*}\begin{array}{rcl} & 6.5 \\ \times & \underline{\;\;\;\;\; 3\;\;} \\ & 19.5 \end{array}\end{align*}

The answer is that Dylan's baby sister will likely weigh approximately 19.5 pounds when she is a year old.

#### Example 2

Estimate the product of

.First, identify the two leading digits of each number. Remember that when the only digit to the left of the decimal point is a 0, it does not count as a leading digit.

- The leading digits of 0.4561 are 0.45
- The leading digits of 0.32109 are 0.32

Now, multiply just as if you were multiplying whole numbers. Because the original numbers have 4 digits total after their decimal points, insert a decimal point into your answer so that it has 4 digits to its right.

\begin{align*}\begin{array}{rcl} & .45 \\ \times & \underline{.32} \\ & \ \ \ 90 \\ + & \underline{1350} \\ & .1440 \end{array}\end{align*}Your estimate is .1440 which is the same as 0.144.

The answer is that the product of 0.4561 and 0.32109 is approximately 0.1440.

**Example 3**

Find the product. Then estimate to confirm your solution.\begin{align*}6.79 \times 1.2\end{align*} .

First, multiply to find the exact answer. You will multiply just as if you were multiplying whole numbers. Because the original numbers have 2 digits total after their decimal points, insert a decimal point into your answer so that it has 2 digits to its right.

\begin{align*}\begin{array}{rcl} & \ \ \ 67.9 \\ \times & \ \ \underline{\;\;\;\; 1.2} \\ & \ \ \ 1358 \\ + & \underline{\;\;\; 6790} \\ & \ \ 81.48 \end{array}\end{align*} Next, estimate the answer by multiplying the leading digits. Identify the two leading digits of each number.

- The leading digits of 67.9 are 67
- The leading digits of 1.2 are 1.2

Now, multiply the leading digits.

\begin{align*}\begin{array}{rcl} & 67 \\ \times & \underline{1.2} \\ & 134 \\ + & \underline{670} \\ & 80.4 \end{array}\end{align*}

Next, compare your answer with your estimate. Your answer of 81.48 is close to your estimate of 80.4. This means you can be confident that your answer is correct.

The answer is that the exact product of

is 81.48, while the estimate of the product is 80.4.#### Example 4

Estimate the product of

.First, identify the two leading digits of each number.

- The leading digits of 5.321 are 5.3
- The leading digits of 2.301 are 2.3

Now, multiply just as if you were multiplying whole numbers. Because the original numbers have 2 digits total after their decimal points, insert a decimal point into your answer so that it has 2 digits to its right.

The answer is that the product of 5.321 and 2.301 is approximately 12.19.

#### Example 5

Find the product. Then estimate to confirm your solution.

.First, multiply to find the exact answer. You will multiply just as if you were multiplying whole numbers. Because the original numbers have 7 digits total after their decimal points, insert a decimal point into your answer so that it has 7 digits to its right.

\begin{align*}\begin{array}{rcl} & \qquad 9.1204 \\ \times & \underline{\;\;\;\;\;\;\;\;\ 8.713} \\ & \qquad 273612 \\ & \ {\;\;\;\;\;\;\; 912040} \\ &{\;\;\;\;\; 63842800} \\ + & \underline{\;\;\ 729632000} \\ & \ \ 79.4660452 \end{array}\end{align*}

Next, estimate the answer by multiplying the leading digits. Identify the two leading digits of each number.

- The leading digits of 9.1204 are 9.1
- The leading digits of 8.713 are 8.7

Now, multiply the leading digits.

Next, compare your answer with your estimate. Your answer of 79.4660452 is close to your estimate of 79.17. This means you can be confident that your answer is correct.

The answer is that the exact product of

is 79.4660452, while the estimate of the product is 79.17.### Review

Estimate the products by multiplying the leading digits.

- \begin{align*}46.14 \times 2.726\end{align*}
- \begin{align*}21.342 \times 14.555\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.11.