2.3 Sub Chapter Notes

Solving a Business Problem • Solving problems involving prices: • Define the variable as the unknown price. • The change in price will then be a percent of the variable. • The variable and its change will equal the known price. Remember: As always, reading the word problem for its facts and relationships is the crucial piece in getting to the answer. What are the facts? What is requested? Will a sketch be useful? Define the variable. Set up the equation. Solve the equation. Check the answer to be sure: • it’s reasonable; • it works arithmetically with the facts; • all the answers asked for have been supplied.

Solving a Mixture Problem • Mixture problems: Word problems that set up equations involving various substances mixed together. • Tricks for mixture problems: • Make sure the amounts are all measured in the same unit. • Convert any percents to two-place decimals. • Remember: the translation process from words to an equation is the true math in word problems and where the real thinking occurs. • In defining the variable, usually one amount is the variable and the other amount involves addition or subtraction with the variable. Read the problem. Draw a sketch. Write the known facts. In this problem: 1. There are 200 g of metal that is 50% silver. 2. This 200 g was made by mixing pure silver with 45% silver metal. Define the variable and create the equation. Read the problem carefully at this point. Remember to change any percents into decimals. Solving this equation is straightforward. Pay careful attention to: 1. arithmetic with the decimals 2. distribution across parentheses.

Solving an Investment Problem • Investment problem: A word problem that requests a solution involving interest on investments.

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Reading the question is always the most important part when solving any word problem. In this example there is $6,000 to invest. Part of the $6,000 is invested at 5%. The remainder is invested at 8.5%. Define the variable, F, as the amount invested at 5%. Then, the amount invested at 8.5% will be ($6,000 – F). The total interest earned is $370. Write an equation where the sum of the interest earned from the two accounts is $370. After converting the percents to decimals, multiply the equation by 1000 to clear the decimals. Then, solve as usual to find the amount invested at 5%. Use the amount invested at 5% and the fact that the total amount invested is $6,000 to find the amount invested at 8.5%.

Solving for Consecutive Numbers • Consecutive numbers: Counting numbers that follow one after another. For example, 6, 7, 8, 9,... are consecutive numbers. Consecutive numbers take the form: x, x + 1, x + 2, … • Consecutive even numbers: Even numbers that follow one another when counting; e.g., 2, 4, 6, .... Consecutive even numbers take the form: 2x, 2x + 2, 2x + 4, … • Consecutive odd numbers: Odd numbers that follow one another when counting; e.g., 5, 7, 9, ... Consecutive odd numbers take the form: 2x + 1, 2x + 3, 2x + 5, … Read the problem. In this example, it’s short and to the point. Determine the facts. Because the requested numbers are odd, they come every other number in the counting sequence. So the problem sets up as the first number, that one plus 2, and that one plus 2 more. Define the variable. Guarantee the numbers are odd by using 2's, which as a multiple of 2 would be even, then add 1 to make the number odd. Write the equation. In this example, it is the sum of the three numbers. Now, solve the equation. NOTE: The variable may not be the answer. Use its value to get the requested information. Remember: Check to see that the requested information has been discovered. Check the answers to see that they work.

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Finding an Average • To find the average: • Add all the terms. • Divide by the number of terms in the total. • To get a desired average, you must: • Add all known terms plus a variable for each unknown item. • Divide the total by the number of items being added. • Set the division problem as equal to the average desired. • Solve for the variable. Step one: read the problem and determine the facts.

Step two: define the variable. In this case the variable is “?” and represents the grade on the test not yet taken. Step three: create the equation. Step four: solve the equation, as usual. Step five: check the answer. Substitute it back into the equation to make it sure it produces a true statement. Finally, circle or box the answer, so it’s easy to spot.

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