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1.1 Problem Solving with Fractions. Addition words Plus, more, more than, added to, increased by, sum, total, sum of, increase of, gain of Subtraction words Less, subtract, subtracted from, difference, less than, fewer, decreased by, loss of, minus, take away.

1.1 Problem Solving with Fractions

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- Addition words
- Plus, more, more than, added to, increased by, sum, total, sum of, increase of, gain of

- Subtraction words
- Less, subtract, subtracted from, difference, less than, fewer, decreased by, loss of, minus, take away

- Multiplication words
- Product, double, triple, times, of, twice, twice as much

- Division words
- Divided by, divided into, quotient, goes into, divide, divided equally per

- Equals
- Is, the same as, equals, equal to, yields, results in, are

- Changing word phrases to expressions:

- Equation: statement that two algebraic expressions are equal.

- Solving Application Problems
- Read and understand the problem
- Know what is given and work out a plan to answer what is to be found.
- Estimate a reasonable answer
- Solve the problem by using the facts given and your plan

- Estimating a reasonable answer: which of the following would be a reasonable cost for a man’s shirt?
- $.65
- $1
- $20
- $1000

- Adding fractions with the same denominator:
- Subtracting fractions with the same denominator:

18

6

3

2

3

- To add or subtract fractions with different denominators - get a common denominator.
- Using the least common denominator:
- Factor both denominators completely
- Multiply the largest number of repeats of each prime factor together to get the LCD
- Multiply the top and bottom of each fraction by the number that produces the LCD in the denominator

- Adding fractions with different denominators:
- Subtracting fractions with different denominators:

- Try these:

- Proper fraction – numerator is less than the denominator
- Improper fraction - numerator is greater than the denominator
- Mixed fraction – sum of a fraction and a whole number

- Converting a mixed fraction to an improper fraction:
- Converting an improper fraction to a mixed fraction:Divide 9 into 35:

- Multiplying or dividing the numerator (top) and the denominator (bottom) of a fraction by the same number does not change the value of a fraction.
- Writing a fraction in lowest terms:
- Factor the top and bottom completely
- Divide the top and bottom by the greatest common factor

- Multiplying fractions:
- Dividing fractions (multiply by the reciprocal):

- Try these:

- Converting decimals fractions:
- Converting fractions to decimals:

- A linear equation in one variable can be written in the form: Ax + B = 0
- Linear equations are solved by getting “x” by itself on one side of the equation
- Addition Property of Equality: if A=B then A+C=B+C
- Multiplication Property of Equality: if A=B and C is non-zero, then AC=BC
- General rule: Whatever you do to one side of the equation, you must also do it to the other side.

- Some equations have more than one term with the same variable. These are called “like terms”
- Like terms can be combined by adding the coefficients:

- Example of solving an equation:

- Translate the following:
- The sum of a number and 16
- Subtract a number from 5.4
- The product of a number and 9
- The quotient of a number and 11
- Four-thirds of a number

- When 5 times a number is added to twice the number, the result is 10. Find the number.
- x is the variable representing the number.
- Equation:
- Solve:
- Check:

I = PRT

M = P(1 + RT)

G = NP

S = C + M

Interest = principal x rate x time

Maturity value

Gross sales = number of items sold x price per item

Selling price = cost of the item + markup

Example: Solve for T in the formula:

Distribute:

Subtract P from both sides

Divide by PR

- Ratio – quotient of two quantities with the same unitsExamples: a to b, a:b, or Note: percents are ratios where the second number is always 100:

- Proportion – statement that two ratios are equalExamples: Cross multiplication:if then

- Solve for x: Cross multiplication:so x = 63

- Write a decimal as a percent by moving the decimal point 2 places to the right and attaching a percent sign:
- Example:

- Write a fraction as a percent by converting the fraction to a decimal and then converting the decimal to a percent:
- Example:

- Write a percent as a decimal by moving the decimal point 2 places to the left and removing the percent sign:
- Example:

- Write a percent as a fraction by first changing the percent to a decimal then changing the decimal to the fraction and reduce:
- Example:

- Write a fractional percent as a decimal by first changing the fractional part to a decimal and leaving the percent sign. Then move the decimal point 2 places to the left and removing the percent sign:
- Example:

- B = Base – the whole or the total
- R = Rate – a number followed by “%” or “percent”
- P = Part – the result of multiplying base times rate

- B = Base – sales, R = Rate – sales tax rate, P = Part – sales tax
- Example: If the sales tax rate is 5%, what is the sales tax and total sale on $133 of merchandise

- Using the Basic Percent Equation to solve for Base:22.5 is 30% of _____

- Finding sales when sales tax rate is given:The 5% sales tax collected by a store was $380. What was the total amount of sales?

- Finding the amount of an investment:The yearly maintenance cost of an apartment is 2½% of its value. If maintenance is $37,000 per year, what is the value of the apartment complex?

- Finding the base if rate and part are different quantities:United Hospital finds that 25% of its employees are men and 720 are women are women. What is the total number of employees?First – if 25% are men, then the percent of women = 100-25 = 75%

- Using the percent equation to solve for rate:45 is what percent of 180?Note: Rate is always expressed as a percent

- Finding rate of return when the amount of return and the investment are known:$3400 is invested in a new computer yielding additional income of $1700. What is the rate of return?

- Solving for the percent remaining:A car is expected to last 10 years before it needs replacement. If the car is 7 years old, what percent of the car’s life remains?To find the number of years remaining subtract 7 from 10 to get 3 years left.

- Find the percent of increase/decrease:Sales of digital cameras went from $40,000 to $100,000. Find the percent increase.Increase = $100,000 - $40,000 = $60,000

- Increase Problem:Original + Increase = New value (base) (part)
- Decrease Problem:Original - Decrease = New value (base) (part)

- The value of a house is $143,000 this year. That is 10% more than last year’s value. What was the value of the home last year?Last year’s value + 10% of last year’s value = this year’s value

- Finding the base after 2 increases:This year’s production of widgets was 144,000. It is 20% more than last year’s production which was also 20% more than the previous year’s production. Find the number of widgets produced 2 years ago. To find last year’s # of widgets:

- Widget problem (continued). To get the # of widgets produced 2 years ago:

- Decrease problem:Craig paid $450 for an LCD TV set. The price he paid was 10% less than the original price. What was the original price?