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# Systems - Word Problems

Lesson Plans > Mathematics > Algebra > Systems of Equations

## Slide Show

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The sum of two numbers is eleven less than four times the smaller number. The difference between the numbers is five more than the smaller number. What is the larger number?

Twenty years ago, Kent was twice as old as Xavier is now. The sum of Kent's age and Xavier's age is 53. How old is each person?

Joni and Edgar go shopping. She buys 3 baseballs and two baseball bats for \$75.00. Edgar buys 5 baseballs and three bats for \$116. How much does a baseball bat cost?

Toby has both nickels and dimes, worth a total of \$3.90. He has a total of 45 coins in all. How many of each does he have?

## Systems - Word Problems

In a previous lesson, students solved systems of equations in which one equation was already solved for one variable, making it very easy to solve using the substitution method. The problems in this lesson require a little more work; most of these problems students will have to rearrange terms to do either addition/subtraction method or substitution method.

Age at Different Times: This problem requires students to think about comparing an age that someone was in the past, with the age someone else is in the present. This can be a tricky concept for some students to wrap their brains around. I find it helpful to make them assign a separate variable for Kent's age 20 years ago (something like Z = Kent's age 20 years ago). It's true that they'll almost immediately write Z = K - 20, but for some students it's easier to have that extra variable to later substitute out.

Collection of Coins: A very standard type of problem in which students have to deal with both the number of coins and the value of coins. Again, it may be helpful to have separate variables for value of each. For example: n = the number of nickels, and N = the value of the nickels. Then they'll have an equation: N = 0.05N. If the problem is done in cents instead of dollars, the decimals can be avoided.

Worksheet: The following may be helpful to point out when students are working on the worksheet

1. In #3, "Thirty-five years ago, Red was twice as old as Uri." Should be interpreted as "Thirty-five years ago, Red was twice as old as Uri was at that time," rather than,  "Thirty-five years ago, Red was twice as old as Uri is now." If the problem was intended to indicate "today" that would imply two different time frames, and would therefore need to be explicitly stated. Without an explicit statement, we assume both sides of the equation are in the same time frame.
2. In #8, we have an actual amount hiked by hiker #1, an actual amount hiked by hiker #2. In addition, there is a "hypothetical" amount hiked by the second hiker. The hypothetical is indicated by the phrase "would have hiked." Students may wish to assign three variables: A, and B for the hikers' mileages, and H for the hypothetical mileage of the second hiker.
3. An important issue arises in #9, which is similar to the decision of cents vs. dollars in the coin problem; we must decide whether we're talking about the cost of an egg, or the cost of a dozen eggs. We can do either, but we must be consistent throughout the problem.
Lesson by Mr. Twitchell

## Sum and Difference

The sum of two numbers is eleven less than four times the smaller number. The difference between the numbers is five more than the smaller number. What is the larger number?

## Ages at Different Times

Twenty years ago, Kent was twice as old as Xavier is now. The sum of Kent's age and Xavier's age is 53. How old is each person?

## Bats and Balls

Joni and Edgar go shopping. She buys 3 baseballs and two baseball bats for \$75.00. Edgar buys 5 baseballs and three bats for \$116. How much does a baseball bat cost?

## Collection of Coins

Toby has both nickels and dimes, worth a total of \$3.90. He has a total of 45 coins in all. How many of each does he have?

## Solutions

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## Word Problem Practice

Pick a variable for each unknown, and state what it represents.  Write two equations and solve.

Example:      The price of two coffees and three teas is \$10.23. The price of one coffee and one tea is \$4.14. What is the price of a coffee?

Answer:     C = price of one coffee; T= price of one tea
2C + 3T = 10.23; C + T= 4.14; C = \$2.19

1. The sum of two numbers is 54, and one number is half the other. What is the smaller number?

2. The difference between two numbers is 25. The larger number is ten more than twice the smaller. What is the larger number?

3. The sum of Red’s age and Uri’s age is 115. Thirty-five years ago, Red was twice as old as Uri. How old is Uri now?

4. The sum of two numbers is ten more than twice the smaller number. The difference between the larger and the smaller is five less than the smaller. What are the numbers?

5. Ten years ago, Paulo was seven times as old as Marty. Paulo’s age is ten more than twice Marty’s age. How old is Marty?

6. After Andi went shopping, the change she received back from all her purchases was one tenth of the amount she spent. She started with \$42.79. How much money did she spend?

7. The temperature today is twenty less than twice what it was yesterday. Yesterday’s temperature was 32 degrees less than today’s temperature. What was the temperature yesterday?

8. Two hikers climbed two different mountains. The first hiker hiked twice as many miles as the second hiker would have hiked, if he had hiked three miles less. Together, they hiked 30 miles. How many miles did the first hiker hike?

9. If eggs were discounted by a certain amount per dozen, The cost for 5 dozen eggs would be \$15.25.  If eggs were discounted by twice the amount, the cost for a dozen eggs would be \$2.90. What was the original price of a dozen eggs?

10. Farmer Jones has cows and chickens. Together, he has 82 animals, and the animals have a total of 258 legs. How many cows does he have?

## Word Problem Practice: Answer Key

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