The SAT Math Test: Problem Solving and Data Analysis
Skill 2: Working with Rates, Ratios, Percentages, and Proportions
Lesson 5: Rates and unit rates
On a sunny day, a 50 square meter section of solar panel array can generate an average of 1 kilowatt-hour of energy per hour over a 10-hour period. If an average household consumes 30 kilowatt-hours of energy per day, how large an array would be required to power 1,000 households on sunny days?
A) 1,500 square meters
B) 15,000 square meters
C) 150,000 square meters
D) 15,000,000 square meters
(Medium) This is clearly a “rate problem,” because it includes two “per” quantities. When working with rates, keep two important ideas in mind:
The units for any rate can be translated to give the formula for the rate. For instance, if a word problem includes the fact that “a rocket burns fuel at a rate of 15 kilograms per second,” this fact can be translated into a formula as long as we remember that per means divided by:
Any “rate fact” in a problem can be interpreted as a “conversion factor.” For instance, if “a rocket burns fuel at a rate of 15 kilograms per second,” then in the context of that problem, one second of burning is equivalent to 15 kilograms of fuel being burned. Therefore, as we discussed in Chapter 7, Lesson 4, we are entitled to use either of the following conversion factors in this problem:
Just as we did in Chapter 7, Lesson 4, we can solve this problem by just noticing that it is essentially a conversion problem. The question asks “how large an array (in square meters) would be required to power 1,000 households on sunny days?” So we can treat the problem as a conversion from a particular number of households to a particular number of square meters of solar panels:
Note very carefully how (1) all of the units on the left side of the equation cancel except for “square meters” (which is what we want), and (2) each conversion factor represents an explicit fact mentioned in the problem.
Many rate problems can be easily managed with the “rate pie”:
This is a simple graphical device to organize information in a rate problem. It is simply a way of expressing all three forms of the “rate equation” at once: distance = rate × time; rate = distance/time; and time = distance/rate. For example, if a word problem states that “Maria completes an x-mile bicycle race at an average speed of z miles per hour,” your “rate pie” should look like this:
First, we plug the given values in: x miles goes in for distance, and z miles per hour goes in for rate. Then, as soon as two of the spaces are filled, we simply perform the operation between them (in this case division) and put the result in the final space. In this case, the time Maria took to complete the race was x/z hours.
A water pump for a dredging project can remove 180 gallons of water per minute, but can work only for 3 consecutive hours, at which time it requires 20 minutes of maintenance before it can be brought back online. While it is offline, a smaller pump is used in its place, which can pump 80 gallons per minute. Using this system, what is the least amount of time it would take to pump 35,800 gallons of water?
A) 3 hour 10 minutes
B) 3 hours 15 minutes
C) 3 hours 25 minutes
D) 3 hours 30 minutes
(Hard) If we want to pump out the water as quickly as possible, we want to use the stronger pump for the maximum three hours. To find the total amount of water pumped in that time, we do the conversion:
So after 3 hours, there are still 35,800 − 32,400 = 3,400 gallons left to pump. At that point, the smaller pump must be used for a minimum of 20 minutes, which can pump
which still leaves 3,400 − 1,600 = 1,800 gallons left. Notice that we have already taken 3 hours and 20 minutes, and as yet have not finished pumping. This means that choices (A) and (B) are certainly incorrect. So how long will it take to pump the remaining 1,800 gallons? Now that we can bring the stronger pump online, it will only take 1,800 gallons × (1 minute/180 gallons) = 10 more minutes; therefore, the correct answer is (D).
Although you don’t need to construct a graph of this situation to solve the problem, graphing helps show the overall picture:
Notice that the line has a slope of 180 for the first 180 minutes, then 80 for the next 20 minutes, and then 180 for the next 180 minutes, and crosses the line y = 35,800 at 210 minutes.
In the graph of any linear function, y in terms of x, the slope of the line is equivalent to the unit rate of the function, that is, the rate at which y increases or decreases for every unit increase in x.
Lesson 6: Ratios: part-to-part and part-to-whole
A marathon offers $5,000 in prize money to the top three finishers. If the first-, second-, and third-place prizes are distributed in a ratio of 5:4:1, how much money, in dollars, does the second-place finisher receive?
(Easy) When given a “part-to-part” ratio, such as 5:4:1 (which is of course, really a part-to-part-to-part ratio), it often helps to add up the parts and consider the whole. This prize is divided into 5 + 4 + 1 = 10 equal parts, so the winner gets 5/10 of the prize money, the second-place finisher gets 4/10 of the prize money, and the third-place finisher gets 1/10 of the prize money. The second-place finisher therefore takes home (4/10) × $5,000 = $2,000.
If you are given a part-to-part ratio, it is often helpful to add up the parts and then divide each part by the sum. For instance, if a paint mixture is a 2:5 combination of red and yellow, respectively, the “whole” is 2 + 5 = 7, which means that the mixture is 2/7 red and 5/7 yellow.
Bronze is an alloy (a metallic mixture) consisting of copper and tin. If 50 kg of a bronze alloy of 20% tin and 80% copper is mixed with 70 kg of a bronze alloy of 5% tin and 95% copper, what fraction, by weight, of the combined bronze alloy is tin?
(Medium) The combined alloy will weigh 50 kg + 70 kg = 120 kg. The total weight of the tin comes from the two separate alloys: (0.20)(50) + (0.05)(70) = 10 + 3.5 = 13.5 kg. Therefore the fraction of the combined alloy that is tin is 13.5/120, which simplifies to 9/80.
Exercise Set 2 (Calculator)
If a train travels at a constant rate of 50 miles per hour, how many minutes will it take to travel 90 miles?
Two cars leave the same point simultaneously, going in the same direction along a straight, flat road, one at 35 miles per hour and the other at 50 miles per hour. After how many minutes will the cars be 5 miles apart?
If a $6,000 contribution is divided among charities A, B, and C in a ratio of 8:5:2, respectively, how much more, in dollars, does charity A receive than charity C?
If a car traveling at 60 mph is chasing a car travelling at 50 mph and is ¼ mile behind, how many minutes will it take the first car to catch the second?
A truck’s gas tank can hold 18 gallons. If the tank is 2/3 full and the truck travels for 4 hours at 60 miles per hour until it runs out of gas, what is the efficiency of the truck, in miles per gallon?
A motorcycle has a fuel efficiency of 60 miles per gallon when it is cruising at a speed of 50 miles per hour. How many hours can it travel at 50 miles per hour on a full tank of gas, if its tank can hold 10 gallons?
If the ratio of a to b is 3 to 4, and the ratio of a to c is 5 to 2, what is the ratio of b to c?
A) 3 to 10
B) 3 to 5
C) 5 to 3
D) 10 to 3
A paint mixture consists of a 3:2:11 ratio of red, violet, and white, respectively. How many ounces of violet are needed to make 256 ounces of this mixture?
A pool that holds 20,000 gallons is ¼ full. A pump can deliver g gallons of water every m minutes. If the pumping company charges d dollars per minute, how much will it cost, in dollars, to fill the pool?
Yael travels to work at an average speed of 40 miles per hour and returns home by the same route at 24 miles per hour. If the total time for the round trip is 2 hours, how many miles is her trip to work?
A hare runs at a constant rate of a miles per hour, and a tortoise runs at a constant rate of b miles per hour, where 0 < b < a. How many more hours will it take the tortoise to finish a race of d miles than the hare?
Janice can edit 700 words per minute and Edward can edit 500 words per minute. If each page of text contains 800 words, how many pages can they edit, working together, in 20 minutes?
If a printer can print 5 pages in 20 seconds, how many pages can it print in 5 minutes?
Traveling at 40 miles per hour, Diego can complete his daily commute in 45 minutes. How many minutes would he save if he traveled at 50 miles per hour?
If and , what is
If a cyclist races at 30 miles per hour for 1/2 of the distance of a race, and 45 miles per hour for the final 1/2 of the distance, what is her average speed, in miles per hour, for the entire race?
Anne can paint a room in 2 hours, and Barbara can paint the same room in 3 hours. If they each work the same rate when they work together as they do alone, how many hours should it take them to paint the same room if they work together?
What is the average speed, in miles per hour, of a sprinter who runs ¼ mile in 45 seconds? (1 hour = 60 seconds)
A car travels d miles in t hours and arrives at its destination 3 hours late. At what average speed, in miles per hour, should the car have gone in order to arrive on time?
In three separate 1-mile races, Ellen finished with times of x minutes, y minutes, and z minutes, respectively. What was her average speed, in miles per hour, for all three races?
Sylvia drove 315 miles and arrived at her destination in 9 hours. If she had driven 10 miles per hour faster, how many hours would she have saved on the trip?
A) 1.75 hours
B) 2.00 hours
C) 2.25 hours
D) 2.50 hours
EXERCISE SET 2 ANSWER KEY
1. 108 time = distance/rate = 90 miles/50 mph = 1.8 hours = 1.8 hour × 60 min/hour = 108 minutes.
2. 20 The fast car is moving ahead of the slow car at a rate of 50 − 35 = 15 mph, and so it will be 5 miles ahead after 5 ÷ 15 = 1/3 hour = 20 minutes.
3. 2,400 Since 8 + 5 + 2 = 15, charity A receives 8/15 of the contribution, and charity C receives 2/15. The difference is 6/15, or 2/5, of the total, which is (2/5)($6,000) = $2,400.
4. 1.5 Since the faster car is catching up to the slower car at 60 − 50 = 10 mph, it will take (1/4 mile)/(10 mph) = 1/40 hours = 60/40 minutes = 1.5 minutes.
5. 20 The tank contains (2/3)(18) = 12 gallons, and travels (4 hours)(60 mph) = 240 miles, so its efficiency is 240/12 = 20 miles per gallon.
6. 12 With 10 gallons of gas and an efficiency of 60 miles per gallon, the car can travel 10 × 60 = 600 miles. At 50 miles an hour this would take 600/50 = 12 hours.
8. A According to the ratio, the mixture is 2/(3 + 2 + 11) = 2/16 = 1/8 violet. Therefore 256 ounces of the mixture would contain (1/8)(256) = 32 ounces of violet paint.
9. C If the pool is ¼ full, it requires (3/4)(20,000) = 15,000 more gallons.
10. B Let x = the distance, in miles, from home to work. Since time = distance/rate, it takes Yael x/40 hours to get to work and x/24 hours to get home.
Multiply by 15:
x = 30 miles
11. B The tortoise would take d/b hours to complete the race, and the hare would take d/a hours to complete the race, so the tortoise would take hours longer.
12. 30 Together they can edit 700 + 500 = 1,200 words per minute, so in 20 minutes they can edit
13. 75 If the printer can print 5 pages in 20 seconds, it can print 15 pages in 1 minute, and therefore 15 × 5 = 75 pages in 5 minutes.
14. 9 Since 45 minutes is ¾ hour, Diego’s daily commute is 40 × ¾ = 30 miles. If he traveled at 50 mph it would take him 30/50 = 3/5 hours = 36 minutes, so he would save 45 − 36 = 9 minutes.
15. 3/10 or 0.3
Multiply by ¾:
16. 36 Pick a convenient length for the race, such as 180 miles (which is a multiple of both 30 and 45). The first half of the race would therefore be 90 miles, which would take 90 miles ÷ 30 mph = 3 hours, and the second half would take 90 miles ÷ 45 mph = 2 hours. Therefore, the entire race would take 3 + 2 = 5 hours, and the cyclist’s average speed would therefore be 180 miles ÷ 5 hours = 36 miles per hour.
17. 1.2 or 6/5 Anne’s rate is 1/2 room per hour, and Barbara’s rate is 1/3 room per hour, so together their rate is 1/3 + 1/2 = 5/6 room per hour. Therefore, painting one room should take (1 room)/(5/6 room per hour) = 6/5 hours.
19. B In order to arrive on time, it would have to travel the d miles in t − 3 hours, which would require a speed of d/(t − 3) mph.
21. B Sylvia traveled at 315/9 = 35 miles per hour. If she had traveled at 35 + 10 = 45 miles per hour, she would have arrived in 315/45 = 7 hours, thereby saving 2 hours.
Lesson 7: Interpreting percent problems
What number is 5 percent of 36?
When interpreting word problems, remember that statements about quantities can usually be translated into equations or inequalities. Here’s a simple translation key:
(Easy) Notice that this enables us to translate the question into an equation, which can be solved to get the answer:
28 is what percent of 70?
Again, let’s use the glossary to translate and then solve:
Divide by 0.7:
40 = x
What number is 120% greater than 50?
To increase a number by x%, just multiply it by . To decrease a number by x%, just multiply it by . For instance, to increase a number by 20%, just multiply by 1.20 (because the final quantity is 120% of the original quantity, and to decrease a number by 20%, just multiply by 0.80 (because the final quantity is 80% of the original quantity).
(Easy) If we increase a number by 120%, the resulting number is 100% + 120% = 220% of the original number. Therefore, the number that is 120% greater than 50 is 2.20 × 50 = 110.
Lesson 8: Percent change
A shirt has a marked retail price of $80, but is on sale at a 20% discount. If a customer has a coupon for 10% off of the sale price, and if the sales tax is 5%, what is the final price of this shirt, including all discounts and tax?
(Medium) To find the final price, we must perform three changes: decrease by 20%, decrease by 10%, and increase by 5%. This gives us (1.05)(0.90)(0.80)($80) = $60.48, so the answer is (C). Notice that, since multiplication is commutative, it doesn’t matter in what order we perform the three changes; the result will still be the same.
If a population of bacteria increases from 100 to 250, what is the percent increase in this population?
To find the percent change in a quantity, just use the formula
Notice that any “percent change” is a “percent of the initial amount,” which explains why the initial amount s the value in the denominator.
(Easy) If we know this formula, this question is straightforward: the percent change is (250 − 100)/100 × 100% = 150%, choice (C). If you mistakenly use 250 as the denominator, you would get an answer of (A) 60%, which is incorrect.
Lesson 9: Working with proportions and scales
On a scale blueprint, the drawing of a rectangular patio has dimensions 5 cm by 7.5 cm. If the longer side of the actual patio measures 21 feet, what is the area, in square feet, of the actual patio?
A) 157.5 square feet
B) 294.0 square feet
C) 356.5 square feet
D) 442.0 square feet
(Medium) In a scale drawing, all lengths are proportional to the corresponding lengths in real life. That is, the lengths in the drawing and the corresponding lengths in real life should all be in the same ratio. We can set up a proportion here to find the shorter side of the patio, x.
When working with proportions, remember the two laws of proportions.
The Law of Cross-Multiplication
If two ratios are equal, then their “cross-products” must also be equal.
The Law of Cross-Swapping
If two ratios are equal, then their “cross-swapped” ratios must also be equal.
7.5x = 105
Divide by 7.5:
x = 14
Therefore, the patio has dimensions 21 feet by 14 feet, and so it has an area of (21)(14) = 294 square feet. The correct answer is (B).
Exercise Set 3 (Calculator)
What number is 150% of 30?
If the areas of two circles are in the ratio of 4:9, the circumference of the larger circle is how many times the circumference of the smaller circle?
What number is 30% less than 70?
What number is the same percent of 36 as 5 is of 24?
David’s motorcycle uses 2/5 of a gallon of gasoline to travel 8 miles. At this rate, how many miles can it travel on 5 gallons of gasoline?
The retail price of a shirt is $60, but it is on sale at a 20% discount and you have an additional 20% off coupon. If there is also a 5% sales tax, is the final cost of the shirt?
If the price of a house increased from $40,000 to $120,000, what is the percent increase in price?
At a student meeting, the ratio of athletes to non athletes is 3:2, and among the athletes the ratio of males to females is 3:5. What percent of the students at this meeting are female athletes?
To make a certain purple dye, red dye and blue dye are mixed in a ratio of 3:4. To make a certain orange dye, red dye and yellow dye are mixed in a ratio of 3:2. If equal amounts of the purple and orange dye are mixed, what fraction of the new mixture is red dye?
If the price of a stock declined by 30% in one year and increased by 80% the next year, by what percent did the price increase over the two-year period?
A farmer has an annual budget of $1,200 for barley seed, with which he can plant 30 acres of barley. If next year the cost per pound of the seed is projected to decrease by 20%, how many acres will he be able to afford to plant next year on the same budget?
If x is % of 90, what is the value of – x?
If n is 300% less than , what is the value of |n|?
The cost of a pack of batteries, after a 5% sales tax, is $8.40. What was the price before tax, in dollars?
If the price of a sweater is marked down from $80 to $68, what is the percent discount? (Ignore the % symbol when gridding.)
Three numbers, a, b, and c, are all positive. If b is 30% greater than a, and c is 40% greater than b, what is the value of ?
If the width of a rectangle decreases by 20%, by what percent must the length increase in order for the total area of the rectangle to double? (Ignore the % symbol when gridding.)
Two middle school classes take a vote on the destination for a class trip. Class A has 25 students, 56% of whom voted to go to St. Louis. Class B has n students, 60% of whom voted to go to St. Louis. If 57.5% of the two classes combined voted to go to St. Louis, what is the value of n?
If 12 ounces of a 30% salt solution are mixed with 24 ounces of a 60% salt solution, what is the percent concentration of salt in the mixture?
If the length of a rectangle is doubled but its width is decreased by 10%, by what percent does its area increase?
The freshman class at Hillside High School has 45 more girls than boys. If the class has n boys, what percent of the freshman class are girls?
If the population of town B is 50% greater than the population of town A, and the population of town C is 20% greater than the population of town A, then what percent greater is the population of town B than the population of town C?
EXERCISE SET 3 ANSWER KEY
1. 45 1.50 × 30 = 45
2. 1.5 Imagine that the areas are 4π and 9π. Since the area of a circle is πr2, their radii are 2 and 3, and their circumferences are 2(2)π = 4π and 2(3)π = 6π, and 6π ÷ 4π = 1.5.
3. 49 70 − 0.30(70) = 0.70(70) = 49.
24x = 180
Divide by 24:
x = 7.5
Multiply by 5/2:
x = 100
6. D 1.05 × 0.80 × 0.80 × $60 = $40.32
7. C (120,000 − 40,000)/40,000 × 100% = 200%
8. D The fraction of students who are athletes is 3/(2 + 3) = 3/5, and the fraction of these who are females is 5/(3 + 5) = 5/8. Therefore the portion who are female athletes is 3/5 × 5/8 = 3/8 = 37.5%.
9. C The purple dye is 3/(3 + 4) = 3/7 red, and the orange dye is 3/(3 + 2) = 3/5 red. Therefore, a half-purple, half-orange dye is (1/2)(3/7) + (1/2)(3/5) = 3/14 + 3/10 = 18/35 red.
10. B If the price of the stock were originally, say, $100, then after this two-year period its price would be (0.70)(1.80)($100) = $126, which is a 26% increase.
11. D The quantity of barley seed is proportional to the acreage it can cover. The cost of seed for each acre of barley was originally $1,200/30 = $40 per acre. The next year, after the 20% decrease, the price would be (0.80)($40) = $32 per acre. With the same budget, the farmer can therefore plant 1,200/32 = 37.5 acres of barley.
12. 1/15 or 0.067 or 0.066
14. 8.00 Let x be the price before tax:
1.05x = $8.40
Divide by 1.05:
x = $8.00
15. 15 (68 − 80)/80 = −0.15
16. 1.82 b = 1.30a and c = 1.40b, so c = 1.40(1.30a) = 1.82a. Therefore c/a = 1.82a/a = 1.82.
17. 150 For convenience, pick the dimensions of the rectangle to be 10 and 10. (This is of course a square, but remember that a square is a rectangle!) This means that the original area is 10 × 10 = 100. If the width decreases by 20%, the new width is (0.80)(10) = 8. Let the new length be x. Since the new rectangle has double the area, 8x = 200, and so x = 25. This is an increase of (25 − 10)/10 × 100% = 150%.
18. 15 The total number of “St. Louis votes” can be expressed in two ways, so we can set up an equation to solve for n:
(0.56)(25) + (0.60)n = 0.575(25 + n)
14 + 0.6n = 14.375 + 0.575n
Subtract 14 and .575n:
0.025n = 0.375
Divide by .025:
n = 15
19. C The total amount of salt in the mixture is (.30)(12) + (.60)(24) = 18, and the total weight of the mixture is 12 + 24 = 36 ounces, so the percent salt is 18/36 = 50%.
20. A If the original dimensions are w and l, the original area is wl. If the length is doubled and the width decreased by 10%, the new area is (0.9l)(2w) = 1.8wl, which is an increase of 80%.
21. C The number of girls in the class is n + 45, and the total number of students is n + n + 45, so the percent of girls is .
22. B B is 50% greater than A:
B = 1.5A
C is 20% greater than A:
C = 1.2A
Divide by 1.2:
B = 1.25C