Sylvan Learning

Solve a multi-step problem involving decimal numbers

Objectives

Student will be able to:

use order of operations to solve a problem involving decimal numbers.
solve a multi-step word problem involving decimal numbers.

Introduction

Margie has two one-dollar bills, five quarters, three dimes, and a nickel in her pocket. How much money does Margie have?

Several steps are necessary to solve this problem.

In order to determine the value of the money in Margie's pocket, first write an expression that represents this amount.

Margie has two one-dollar bills, and each dollar bill is worth $1.00, so the expression that represents the value of the dollar bills is $\inline 2\times 1.$

Margie has five quarters, and each quarter is worth $0.25, so the expression that represents the value of the quarters is $\inline 5\times 0.25.$

The expression that represents the value of the three dimes is $\inline 3\times 0.10,$ and the expression that represents the value of the nickel is 0.05.

Now, write the sum of the values.

$\left ( 2\times 1.00 \right )+\left ( 5\times 0.25 \right )+\left ( 3\times 0.10 \right )+0.05$

This expression represents the value of the money in Margie's pocket. Notice that the expression contains more than one operation.

Use order of operations to simplify the expression. Order of operations is a set of steps to use when simplifying expressions that contain more than one operation. Order of operations insures that only one value exists for a given expression.

**Order of Operations**
Step 1:	Simplify parentheses and other grouping symbols.
Step 2:	Multiply and divide, left to right.
Step 3:	Add and subtract, left to right.

First, simplify parentheses. Remember to count the decimal places in the factors to determine the decimal placement in the product.

2.00+1.25+0.30+0.05

Next, add to find the sum. Remember to line up the decimal points.

$\begin{align*}2.00\\ 1.25 \\ 0.30\\ \underline {+~~0.05}\\ 3.60\end{align*}$

Margie has a total of $3.60.

Examples

Use order of operations to simplify the following expression.

$2\left ( 5.7-1.6 \right )+3\times 4.2$

Step 1: Simplify parentheses.

$2\left ( 4.1 \right )+3\times 4.2$

Step 2: Multiply from left to right.

Step 3: Add.

20.8

Three brothers are saving money to buy a game table. Paul has $27.38 saved. James has twice as much as Paul, and Matt has $15.17 less than Paul. Do they have enough saved to purchase a game table for $100.00?

First, write an expression to represent the total amount the brothers have saved.

Paul has $27.38.

James has $\inline 2\times \$ 27.38.$

Matt has $\inline \$ 27.38-\$ 15.17.$

The sum of these expressions will represent the total amount the brothers have saved.

$27.38+\left ( 2\times 27.38 \right )+\left ( 27.38-15.17 \right )$

Use order of operations to simplify the expression.

What is the first step?

The first step is to simplify the parentheses.

27.38+54.76+12.21

What step is next?

Since addition is the only operation left, add the decimal numbers, remembering to line up the decimal points.

$\begin{align*}27.38\\ 54.76 \\ \underline {+~~12.21}\\ 94.35\end{align*}$

The brothers have a total of $94.35 saved, so they do not have enough to purchase a $100.00 game table.

Two students simplify the expression below.

$3.2\times 4.8\div 1.6\times 3.75.$

Susan thinks the answer is 36, but Jack thinks the answer is 2.56. Which student is correct?

To check, simplify the expression using the order of operations.

Since there are no parentheses, multiply and divide. Remember that multiplication and division are completed in the same step, so work from left to right.

$3.2\times 4.8\div 1.6\times 3.75$

$15.36\div 1.6\times 3.75$

$9.6\times 3.75$

The expression simplifies to 36, so Susan is correct.

Determine what Jack did wrong.

Jack failed to work from left to right. He incorrectly multiplied $\inline 1.6\times 3.75$ first, which is at the far right of the problem. This makes the division by 6, rather than by 1.6 as the problem indicates.

Remember: Multiplication and division are done in the same step, working from left to right across the problem. In the final step, the same holds true for addition and subtraction.

Try Together

Complete the exercises. Show all work.

1. Simplify. $2.5\left ( 64.73-29.16 \right )+8.4\div 2.1$

92.925 Check student's work.

2. Paula has 17 quarters, 23 dimes, 5 nickels, and 44 pennies. What is the value of Paula's coins?

Paula has $7.24. Check student's work.

3. A rectangular swimming pool has a length of 45.23 feet and a width that is one-half of the length. A vinyl cover is being made to help keep grass and leaves out of the water. The cover will extend 3 feet on each side in order to stay anchored to the ground. Find the area of the vinyl cover. Round the answer to the nearest hundredth. (Hint: Area = length x width)

The area of the vinyl cover is approximately 1,465.95 square feet. $\inline A=\left ( 45.23+3+3 \right )\times \left ( 45.23\div 2+3+3 \right )$ Check student's work.

Please evaluate the student's answers:

Excellent

Good

Okay