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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?