QR ch.11
Ratio
- A ratio is the proportional relationship between 2 quantities. It may be expressed with a colon (2:3), in words ( the ratio of 2 to 3) or as a fraction (2/3). (Note 2/3 chocolate to glazed donuts would mean that there are 2/5 chocolate and 3/5 glazed.)
Comparing positive fractions
- Given 2 positive fractions with the same denominator, the fraction with the larger numerator will have the greatest value. - Given 2 positive fractions with the same numerator but different denominators, the fraction with the smaller denominator will have the largest value. Ex: 3/4 or 3/8? 3/4. - If neither are the same, you can: 1. Convert to decimal form 2. Express as equivalent fractions 3. Cross multiply (ex: 5/6 or 7/9? 45>42)
Percent Word Problems
- Profit made on an item = sellers price minus cost to the seller. ex: profit / original selling price (100%) = percent of selling price that is profit. - Discount is the original price minus the reduced price. ex: Percent Discount = Discount/ Original Price (100%) - A sales price is the final price after discount or decrease. - Interest is a percent per unit of time.
Combined Rate Problems
- Rates can be added. ex: Nelson moves 200 square meters per hour, John can mow 100 square meters per hour. How long will it take to mow 1800 square meters? 200 + 100 = 1800/300 = 6 hours.
Properties of 100%
- The key to solving GRE percent problems is to realize that all parts add up to one whole, 100%.
Converting Decimals
- To convert a decimal between 0 and 1 to a fraction, determine the place value of the last nonzero digit and set that value as the denominator. Then use all the digits of the decimal number as the numerator, reduce the fraction. ex: Convert 0.875 to a fraction. 875/1000 = 7/8. - To convert a fraction to a decimal, just divide the numerator by the denominator. 4/5= 0.8.
Using the percent formula to solve percent problems.
- You can solve most percent problems by plugging the given data into the percent formula: Part/Whole(100%) = Percent. ex: 1/5(100%) = 20%
Properties of Fractions between -1 and 1
-The reciprocal of a fraction between 0 and 1 is greater than both the original factor and one. ex: 1/2= 2/1 ; 2/1= 2 > 1/2 & 1 . - The reciprocal of a fraction between -1 and 0 is less than both the original fraction and negative one. ex: -1/4 = -4/1 = -4 ; -4 < -1/4 & -1. - The square of a fraction between 0 and 1 is less than the original fraction. ex: (1/2)^2 = (1/2)(1/2)= 1/4. 1/2 > 1/4. - The square of any fraction between 0 and -1 is greater than the original fraction. ex: (-1/2)^2 = (-1/2)(-1/2)= (1/4). -Multiplying any positive number by a fraction between 0 and 1 gives a product smaller than the original number. ex: 6(1/4) = 6/4 = 3/2. - Multiplying any negative number by a fraction between 0 and 1 gives a product greater than the original number. ex: (-3)(1/2)= -3/2
To convert an improper fraction to a mixed number
Divide the numerator by the denominator. The amount of times the denominator fully goes into the numerator is the integer, the remainder is the fraction.
Dividing Fractions
Invert (reciprocal) the second fraction and multiply Ex: (1/2) / (3/5) = (1/2) x (5/3) = 5/6
To convert a mixed number to a fraction
Multiply integer by the denominator and add the numerator. This new number is your numerator, denominator doesn't change. Ex: 2(3/7) = 7(2)+3 / 7 = 17/7
Picking Numbers
-Always pick the number 100 to substitute in percent values, because its very easy to find percentages of 100.
Comparing decimals
Knowing place values allows you to assess the relative values of decimals. ex: 0.254 or 0.3? 0.3 = 3/10 = 300/1000. 0.254 = 254/1000. 300>254.
Multistep Percent Problems
- You cannot add percents of different wholes.
Combining Averages
You can combine averages when there is an equal number of terms in each set. ex: Average = Sum of Terms/ Number or Terms = x(y) + x(z)/ n A weighted average occurs when the number of terms in the set is different.
improper fraction
a fraction whose numerator is larger than the denominator.
Mixed Number
a number made up of an integer and a fraction.
Converting Percents
-To change a fraction to its percent equivalent, multiply by 100%. ex: What is the percent equivalent of 5/8? 5/8(100)= 500%/8 = 62(1/2). - To change a decimal fraction to a percent, you can use the rules for multiplying by powers of 10. Move the decimals two places to the right and insert a percent sign. ex: what is the percent equivalent of 0.17? 17%. - To change a percent to its fractional equivalent, divide by 100 %. ex: Fraction equivalent of 32%? 32/100= 8/25. - To convert a percent to its decimal equivalent, use the rules for dividing by powers of 10. Just move the decimal place two places to the left. ex: The decimal equivalent of 32%? 0.32. - When you divide a percent by another percent, the percent sign "drops out" just as you would cancel out a common factor. 100%/5% = 100/5 = 20. (There are 20 groups of 5% in 100%) -When you divide a percent by a regular number, the percent sign remains. ex: 100%/5 = 20% ( One-fifth of 100% is 20%.)
Combined Work Formula
1 job units per "a" amount of time. T is the time it takes all three people to do the job 1/a + 1/b + 1/c = 1/T. or T = ab/ a+b or T= abc/ ab + bc+ ac .
Common Percent Equivalents
1/20= 5% , 1/2 = 50% 1/12 = 8 1/3% , 3/5 = 60% 1/10 = 10% , 5/8 = 62 1/2 % 1/8 = 12 1/2 , 2/3 = 66 2/3 % 1/6 = 16 2/3 % , 7/10 = 70 %
Fraction Word Problems
Part/Whole = Fraction Part/Whole = Decimal Part/Whole(100) = Percent - To avoid careless errors, look for the key words is and of. Is (or are) often introduce the part, where of introduces the whole.
Percent Increase and Decrease
Percent Increase = Increase(100%) / Original Percent Decrease = Decrease (100%) / Original. ex: 150/450(100%) = 33 1/3 %
Speed Formula
Speed = Distance / Time Time = Distance / Speed Distance = Speed x Time
Using the Average to Find a Missing Number
Sum of Terms = Average x Number of Terms
Other Rates
Time = Quantity/ Rate
Multiplying Fractions
To multiply fractions, multiply the numerators and multiply the denominators Ex: (5/7)(3/4)= 15/28