Ratios and Proportions

Words of “Wiz-dom”-Ratio problems are just fractions and percents in disguise. Think about them as parts of a whole. Just like 3/4 means three of four parts with four parts making up the whole, a ratio of 3:1 means the same thing. It means three parts to one part with four parts as the whole.

The test writer wants to know if you realize the key to ratio problems is the sum of the parts.

Let’s say we are talking about a bag that contains red marbles and green marbles.  The SAT writers will sometimes say “a ratio of 3/2 (and use a horizontal slash) ” red marbles to green marbles, which means the same thing as 3:2, red:green.  In both cases, you always add up the parts, in this case you get 5 as the size of the group.  Then convert the ratio to  3/5 and 2/5 respectively.  Notice the numbers in the ratio are now the numerators of fractions with the sum of the ratio as the denominator.  You may also convert the fractions to decimals (0.6 and 0.4) if that makes it easier to use your calculator.

 Keep in mind the denominator or sum of the parts of the ratio is the size of the group, which may be the total population but most likely it is not.  The total population will be a multiple of the group.  For example, if there are 20 marbles in the population the ratio is 12:8, which reduces to 3:2, the size of the group.  In this example, there cannot be 27 in the population.  Why?  The sum (5) does not divide evenly into 27.  That means that  and  of 27 end up as parts of a member of the population (16.2 and 12.8) and you can’t have 0.2 or 0.8 of a marble!

Sample Questions

5.  Ms. Wiz is going to give students 120 bonus points for completing an optional science project in my lab. Nick, Brandon, and Kristen worked together to complete the project. Nick put in 60 hours, Brandon put in 45 hours and Kristen put in 75 hours. If they split the bonus points based on the hours they each worked, rounded to the nearest whole number, how many bonus points should Kristen get?

A. 50
B. 60
C. 75
D. 90
E. 120

Answer to question #5

10.  The ratio of left-handed to right-handed students is 1:19 at Much Wizer High. If there are 1800 students at the school, how many are left-handed?

(A) 18
(B) 36
(C) 90
(D) 1700
(E) 1710

Answer to Question #10

12.  The ratio of juniors to seniors is 21 to 19 at Draw Bridge High School. How many students could be juniors and seniors combined?

(A) 1250
(B) 1420
(C) 1580
(D) 1600
(E) 1650

Answer to Question #12

9.  The ratio of seniors to juniors to sophomores who bought yearbooks at Dragon Breath High is 11:9:5. What percent of the yearbook buyers were seniors?

(A) 11
(B) 20
(C) 40
(D) 44
(E) 72

Answer to Question #9

Proportions

Proportion questions may ask about directly or indirectly proportional.  If directly proportional, as one thing goes up so does the other; you can set up an equation that allows you to cross multiply.  Be sure you keep the units in the corresponding position.  For example in question 10, make sure you set it up so that when you cross multiply, the solution comes out in inches.  If indirectly proportional, as one thing goes up the other goes down;  You set up an equation in which the right side of the equation is inverted (indirect) so that the two conditions are arranged as follows: =.  Make sure the units in the numerator and denominator are the same.  For example, if driver A drives 60mph for 2 hours, how long does it take driver B get who drives 45mph to drive the same distance?  Set it up as  =  and cross multiply.  The answer is 2  hours.  Be sure you look at your answer in proportion problems and see if it makes sense.

7.  The window cranks in the Wizard’s castle raise the window 9 inches every time they are turned twice.  How high will the window be after turning the crank eight times?

(A) 18
(B) 36
(C) 72
(D) 90
(E) 120

Answer to question #7

4.  A model castle company sells models that are on a scale of 24:1. If an original castle tower was 84 feet tall, how many inches tall will the model tower be?

(A) 3
(B) 3.5
(C) 36
(D) 42
(E) 84

Answer to question #4

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