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Look at the picture of the bananas and the information about the price of bananas below. Can you figure out how much just one banana would cost? In this concept, we will use proportions and logical reasoning to determine the cost of individual items given information about multiples of those items.

Guidance

In order to answer a question about the cost of an item like the one above, we can use the problem solving steps to help.

  • First, describe what you know. What do we know about the weight of the fruit? What do we know about the cost of the fruit?
  • Second, identify what your job is. In these problems, your job will be to figure out the cost or weight of one piece/pound of fruit. Or, it might be to figure out how many pieces of fruit are in a closed box.
  • Third, make a plan . In these problems, your plan should be to write a proportion that relates the weight of the fruit to the cost. Solve the proportion and use logical reasoning to answer the question that was asked.
  • Fourth, solve . Implement your plan.
  • Fifth, check . Check your answer by verifying that it works with all of the original information.

Example A

Look at the pictures below and answer the question.

Solution:

We can use problem solving steps to help us.

& \mathbf{Describe:} && \text{There are}\ 6\ \text{oranges in a box weighing a total of}\ 1.2\ \text{pounds. A sign shows that}\\& && 1\ \text{pound of oranges cost}\ \$0.69. \\\\& \mathbf{My \ Job:} && \text{Figure out the cost of one orange. Assume oranges weigh the same.} \\\\& \mathbf{Plan:} && \text{Use a proportion to figure out the cost of}\ 1.2\ \text{pounds of oranges}. \\ & && \text{Then divide that cost by 6 to get the cost of one orange}. \\\\& \mathbf{Solve:} && \frac{1\ pounds}{\$ 0.69} = \frac{1.2\ pounds}{x \ dollars}; 1x = \$0.69 \times 1.2.\ \text{So},\ x \approx \$0.83.\\& && \text{One orange costs about}\ \$0.83 \div 6,\ \text{or}\ \$0.14. \\\\& \mathbf{Check:} && \text{One orange weighs}\ 1.2 \div 6,\ \text{or}\ 0.2\ \text{pounds and costs}\ \$0.14. \\& && 6\ \text{oranges} \times 0.2\ \text{pounds/orange} = 1.2\ \text{pounds}. \\& && 6\ \text{oranges} \times \$0.14/\text{orange} = \$0.84

Example B

Look at the pictures below and answer the question.

Solution:

We can use problem solving steps to help us.

& \mathbf{Describe:} && \text{There are}\ 9\ \text{avocados in a box weighing a total of}\ 1.5\ \text{pounds. A sign shows that}\\& && 4\ \text{avocados cost}\ \$2.40. \\\\& \mathbf{My \ Job:} && \text{Figure out the cost of one pound of avocados. Assume avocados weigh the same.} \\\\& \mathbf{Plan:} && \text{Use a proportion to figure out the cost of}\ 9\ \text{avocados}. \\ & && \text{Then divide that by 1.5 to get the cost of one pound}. \\\\& \mathbf{Solve:} && \frac{9 \ avocados}{x \ dollars} = \frac{4 \ avocados}{\$ 2.40}; 4x = \$2.40 \times 9.\ \text{So},\ 4x = \$21.60,\ \text{and}\ \$21.60 \div 4 =  \$5.40. \\& && \text{1 pound of avocados costs}\ \$5.40 \div 1.5,\ \text{or}\ \$3.60. \\\\& \mathbf{Check:} && \text{One avocado weighs}\ 1.5 \div 9,\ \text{or}\ 0.167\ \text{pounds and costs}\ \$0.60. \\& && 9 \ \text{avocados} \times 0.167\ \text{pounds/avocado} = 1.5\ \text{pounds}. \\& && 9\ \text{avocados} \times \$0.60/\text{avocado} = \$5.40

Example C

Look at the pictures below and answer the question.

Solution:

We can use problem solving steps to help us.

& \mathbf{Describe:} && \text{There are}\ 6\ \text{grapefruits in a box costing a total of}\ \$2.10.\ \text{A sign shows that}\\& && 2\ \text{pounds of grapefruits cost}\ \$1.00. \\\\& \mathbf{My \ Job:} && \text{Figure out the weight of one grapefruit. Assume grapefruits weigh the same.} \\\\& \mathbf{Plan:} && \text{Use a proportion to figure out the weight of}\ \$2.10\ \text{worth of grapefruits}. \\ & && \text{Then divide that weight by 6 to get the weight of one grapefruit}. \\\\& \mathbf{Solve:} && \frac{2\ pounds}{\$ 1.00} = \frac{x\ pounds}{\$2.10}; 1x = \$2.10 \times 2.\ \text{So},\ x = 4.2 \ \text{pounds.} \\& && \text{One grapefruit weighs}\ 4.2 \div 6,\ \text{or}\ 0.7 \ \text{pounds.} \\\\& \mathbf{Check:} && \text{One grapefruit costs}\ \$2.10 \div 6,\ \text{or}\ \$0.35\ \text{and weighs}\ 0.7 \ \text{pounds.} \\& && 6\ \text{grapefruits} \times 0.7\ \text{pounds/grapefruit} = 4.2\ \text{pounds}. \\& && 6\ \text{grapefruits} \times \$0.35/\text{grapefruit} = \$2.10

Concept Problem Revisited

We can use problem solving steps to help us.

& \mathbf{Describe:} && \text{There are}\ 8\ \text{bananas in a box weighing a total of}\ 3.2\ \text{pounds. A sign shows that}\\& && 4\ \text{pounds of bananas cost}\ \$2.00. \\\\& \mathbf{My \ Job:} && \text{Figure out the cost of one banana. Assume bananas weigh the same.} \\\\& \mathbf{Plan:} && \text{Use a proportion to figure out the cost of}\ 3.2\ \text{pounds of bananas}. \\ & && \text{Then divide that cost by 8 to get the cost of one banana}. \\\\& \mathbf{Solve:} && \frac{4\ pounds}{\$ 2.00} = \frac{3.2\ pounds}{x \ dollars}; 4x = \$2.00 \times 3.2.\ \text{So},\ 4x = \$6.40,\ \text{and}\ \$6.40 \div 4 =  \$1.60. \\& && \text{One banana costs}\ \$1.60 \div 8,\ \text{or}\ \$0.20. \\\\& \mathbf{Check:} && \text{One banana weighs}\ 3.2 \div 8,\ \text{or}\ 0.4\ \text{pounds and costs}\ \$0.20. \\& && 10\ \text{bananas} \times 0.4\ \text{pounds/banana} = 4\ \text{pounds}. \\& && 10\ \text{bananas} \times \$0.20/\text{banana} = \$2.00

Vocabulary

A proportion is an equation that states that two ratios are equal. In this concept, we can use proportions because the ratio of weight to cost is constant for each type of fruit. As long as we know the cost of fruit for a particular weight of that fruit, we can find the cost of that fruit for any weight of that fruit using proportions.

Guided Practice

1.

2.

3.

Answers:

1. \$0.05; \frac{3\ pounds}{\$ 1.20} = \frac{1.5\ pounds}{x \ dollars} , so x = \$0.60 , and \$0.60 \div 12 = \$0.05 .

2. 10; \frac{3\ pounds}{\$ 0.90} = \frac{4\ pounds}{x \ dollars} , so x is \$1.20 , and \$1.20 \div \$0.12 is 10.

3. 21; \frac{4\ pounds}{\$ 5.04} = \frac{5\ pounds}{x \ dollars} , so x = \$6.30 , and \$6.30 \div \$0.30 = 21

Practice

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Difficulty Level:

At Grade

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Date Created:

Jan 18, 2013

Last Modified:

May 27, 2014
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