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Write an Equation

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Steps in Solving a Word Problem 1.Represent an unknown quantity with a variable. 2.When necessary, represent other conditions in the problem in terms of the variable.

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Steps in Solving a Word Problem 3. Identify two equal quantities in the problem. 4. Write and solve an equation. 5. Check the answer.

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Example 1 The price of a sports coat is reduced by $25.80. The new price is $103.20. What was the original price of the coat?

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Think: What is the unknown in the problem? Let a variable represent the unknown value. Let p = the original price.

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Think: Identify two equal quantities. The original price reduced by $25.80 equals the sale price of $103.20. Write the equation; then solve it. p – 25.8 = 103.2 p – 25.8 + 25.8 = 103.2 + 25.8 p = $129 The original price of the sports coat was $129.

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Think: Does this answer seem to be a reasonable price for a sports coat? Does this answer fit the conditions of the problem? This is a reasonable answer, and fits the conditions of the problem since $129 – $25.80 = $103.20. This is the correct answer.

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The teacher says, “Take any number and add 4 to it. Now multiply the sum by 6. Next subtract 12 from the product. Divide the difference by 6 and subtract 3 from the quotient. Now tell me the answer and I will tell you the original number.” Example 2

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He always gives the correct number. How is he able to do this? Write the series of steps in the problem, using n for the original number. Why does this work?

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Think: Determine what you are trying to find. You want to know what the number is and how the teacher always knows the correct answer.

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Think: The answer depends on the original number. How does the teacher know the original number after the answer to the calculation is found? Is there a constant relationship between the original number and the final answer?

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Try several original numbers and see if there is a constant relationship between the original number and the answer.

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If the original number is 2, then 2 + 4 = 6; 6(6) = 36; 36 – 12 = 24; 24 ÷ 6 = 4; 4 – 3 = 1. If the original number is 5, then 5 + 4 = 9; 9(6) = 54; 54 – 12 = 42; 42 ÷ 6 = 7; 7 – 3 = 4. If the original number is 8, then 8 + 4 = 12; 12(6) = 72; 72 – 12 = 60; 60 ÷ 6 = 10; 10 – 3 = 7.

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Think: What is the relationship between the original number and the final calculation? The final calculation is always 1 less than the original number. So the teacher knows that if he adds 1 to the final calculations, he will have the original number.

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Think: Why does this work? Let the original number be n, and find the algebraic expression that represents the arithmetic calculations. n + 4 6(n + 4) 6(n + 4) – 12 6(n + 4) – 12 6 – 3

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Think: What happens if this expression is simplified? = n + 2 – 3 6(n + 4) – 12 6 – 3 = – 3 6n + 24 – 12 6 6n + 12 6 = – 3 6n 12 = + – 3 66 66 = n – 1

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This shows that no matter the value of n, the result of the calculation will be 1 less than the original number. So if you add 1 to the final result, you will obtain the original number.

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Mrs. Evans made three desserts, each containing a different amount of calories. One serving of the banana pudding had six times as many calories as a cookie, and a doughnut had three times as many calories as a cookie. Example

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If a serving of each had a total of 880 calories, how many calories did each have?

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Example Five teenage boys split the cost of pizza. The total bill plus tax was $38.75. How much did each boy pay?

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Exercise Five people went together to buy pizza costing $18.95. How much did each person pay?

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The amount that the boys collected for the mission trip was four times the amount that the girls collected. Together they collected $124.25. How much did the girls collect? Exercise

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You tell your friend, “Choose any number, subtract 8 from it, multiply the difference by 6, add 48, and then divide the result by 2.” How can you always give the original number? Give an algebraic reason as to why your answer is always correct. Exercise

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The soccer players were required to run 6 more laps per day than the tennis players for five straight days. If the soccer players ran 45 laps total during these days, how many laps did the tennis players run each day? Exercise

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Bodie scored 94%, 82%, and 85% on his first three math tests. What score must he make on the next test to have an overall average of 90%? Exercise

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Miss Sneed has nine more boys than girls in her class. If she has a total of 37 students in her class, how many boys and girls does she have in her class? Exercise

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Stefan collects baseball cards. He has three times as many National League players’ cards as American League players’ cards. If he has 212 cards in his collection, how many of each league does he have? Exercise

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Taylor consumed a total of 580 calories in dessert (cupcakes and cookies). Each cupcake had 140 calories and each cookie had 80 calories. If he ate one more cupcake than cookie, how many of each did he eat? Exercise

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