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Work Backwards

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Example The manager of an appliance store multiplied the cost of a kitchen range by 1.3 to get the selling price, added $35.69 for sales tax, and then added $40 for delivery and installation.

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The resultant selling price to the customer was $789.39. What was the dealer’s cost? Example

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Think: What are we supposed to find in this problem, and what information do we know? We are to find the dealer’s cost. We know the final cost and a series of steps that the dealer used to determine the final cost.

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Think: A diagram might help show the steps that were used to determine the total cost to the buyer. The diagram illustrates the order of the operations as stated in the problem.

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Think: The total cost to the buyer is known after the series of calculations. If we take the total cost to the buyer and “undo” the calculations in reverse order, then the cost to the dealer will be the result.

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Cost of range × 1.3 + 35.69 + 40 789.39 Work backwards using inverse operations to solve the problem. Cost of range ÷ 1.3 – 35.69 – 40 789.39

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789.39 – 40.00 749.39 749.39 – 35.69 713.70 $549 Cost of range ÷ 1.3 – 35.69 – 40 789.39 1.3 713.70 54.9

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The diagram illustrates the working backward process. The inverse operations are used in reverse order.

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Think: Does this answer of $549 seem to be a reasonable answer for this problem? Using $549, check the forward series of calculations and see if the final result is the total cost to the buyer.

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549(1.3) = 713.70 Multiply by 1.3. 713.70 + 35.69 = 749.39 Add sales tax. 749.39 + 40 = 789.39 Add the delivery fee. The answer of $549 is correct.

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The class president budgeted the money allotted for the class party. He budgeted for decorations, for games, and for food. 1414 1414 1515 1515 1313 1313 Example

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If the decorations cost $88, how much was spent on games and food? How much money was left over? Example

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The football team ended their possession at their 45 yd. line. During the possession they gained 22 yd., lost 15, gained back 6, and were penalized 5 yd. Where did they begin this possession? Example

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The manager of an electronics store multiplied the cost of a new TV by 1.5 to get his selling price. He then deducted $50 for a minor scratch, added $41.25 sales tax, and added $49.99 for an extended service plan. Exercise

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The resulting cost to the consumer was $1,278.74. What was the merchant’s cost for the TV?

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From 1970 to 1980 a city’s population increased by 1,450. From 1980 to 1990 its population doubled. From 1990 to 2000 its population decreased by 798 to a population of 16,468. What was the population in 1970? Exercise

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Two less than one-fourth of the soccer team came down with a virus before the big game and were unable to play. If 3 players were unable to play, how many players are on the team? Exercise

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Shanda climbed a set of stairs and stopped on the middle step. She then walked down 8 steps, up 6 steps, and down 10 steps, ending at the bottom of the stairs. How many steps are in the set of stairs? Exercise

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Start with the variable x and perform the following operations on it in the order given. Subtract 9; multiply by 4; add 8; and divide by 12. What would the value of x be if the result is 2? Exercise

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Uncle Larry left half of his estate to be split three ways between you and your two brothers. After this division, half of what was left went to his college, and the remaining $100,000 was left to his church. Exercise

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What was the total value of his estate? How much did you and each of your two brothers get?

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Lincoln Middle School is planning an old-fashioned carnival. The students have to budget their purchases wisely. Exercise

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They decide to spend a third of their budgeted allowance on prizes, a fourth on costumes, and two-fifths on booth construction. If they have $5 of petty cash left after their purchases, what was the budget allotted to them?

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James spent his birthday money at four different places. He spent half of his money on CDs, a fourth of what was left playing paintball, a third of what was left at the video store, and a sixth of the rest on candy. Exercise

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If he had $10 left from his birthday money, what was the total amount that he got for his birthday?

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