Ratios and Unit Rates – Blue Problems
Part 1. Problems Involving RATIOS
1. There are 12 red marbles and 12 black marbles in a jar. What is the least number
of marbles that must be removed so that the ratio of red to black marbles
remaining in the jar is 4 to 3?
2. Express the ratio of 2 to 4 as a decimal to the nearest tenth.
7
5
3. If a : b : c = 3:5:6 and c : d = 1:5 then what is the ration of b to d? Express your
answer as a common fraction.
4. Honeybees can lift 300 times their own weight. If human could lift weight in this
same ratio. How many pounds would a person need to weigh to lift one ton? One ton
equals 2000 pounds. Express your answer as a decimal to the nearest tenth.
5. The ratio of boys to girls in a class of 35 students is 3:4. How many more girls than
boys are in the class?
6. When an amount of money is shared in the ratio 5:11, the smaller share is $20. How
much money was shared?
7. Jodie and Sarah start a business selling their home-made lemonade. To get started,
Jodie invests $20 and Sarah invests $15. They decide to share the profits in the
ratio of their investments. How will they share the first week’s profit of $22.40?
8. For her party, Michelle makes drinks using her special recipe. Orange juice, Lemon
juice and mineral water are mixed in the ratio of 5:2:8.
a. How many milliliters of each will be needed to make 3 liters of drinks?
b. Michelle finds that she only has 250 mL of lemon juice. How much drink will
she be able to make using her recipes?
9. A length of ribbon 2 meters long is to be cut into four pieces in the ratio of
2:3:5:6. Find the length of the longest piece.
10. There are red and blue marbles in a jar. One-third of the marbles are red. If
3
another red marble is added to the jar, then 8 of the marbles are red. How many
blue marbles are in the jar?
Problems Involving Conversions
1. A rental company charges $45 per day and 35¢ per mile to rent a car. What is the
maximum whole number of miles that can be driven in one day and still keep the
cost less than $125?
2. Jessica completed 7/8 of a training run in 2 hours and 55 minutes. If Jessica runs
at a constant pace, what is the number of minutes left in her run?
3. Xa worked 2 hours and 15 minutes. Yo worked 1 hour and 40 minutes, and Zu worked
2 hours and 30 minutes. Combined, they were paid $77. If each of them is paid the
same hourly wage, how many dollars should Xa get?
4. A faucet leaks at a rate of 1 pint every 2 hours for 4 full days. How many gallons of
water leaked from the faucet?
Note: 1 gallon=4quarts and 2 pints =1 quart
5. A car’s odometer read 62,926 miles at 10:18 a.m. At 11:58 a.m., the odometer
displayed the next possible number that read the same forward and backward.
During that time, what was the car’s average speed in miles per hour?
6. On a certain day, 10 US dollars were worth 7.60 Euros, and 1 euro was worth 155
Japanese yen. How many yen were the 10 US dollars worth?
7. A car that is traveling 90 feet per second is traveling how many miles per hour?
There are 5280 feet in 1 mile. Express your answer to the nearest whole number.
8. Paitlyn estimates she will need $750 for the computer she wants. She baby-sits for
the Knox family every Saturday for $5.50 per hour. On average, she baby sits 4.5
hours each Saturday. What is the fewest number of weeks she must work in order
to save enough money for the computer?
9. The ratio of rise to run for a step should be between 0.45 and 0.60. If the rise of
a step is 6¾˝and the run is 13¼˝, find the ratio of rise to run. Express your answer
as a decimal to the nearest hundredth.
10. Howard ran a half-marathon (13.1 miles) in 2 hours. He completed the first 10 miles
in 95 minutes. How many minutes per mile did Howard average for the remainder of
the race? Express your answer as a decimal to the nearest tenth.
11. The speed of light is 670,000,000 miles per hour. How many seconds does it take
light to travel 121,000,000 miles? Express your answer to the nearest second.
1 of a day. How many minutes long is a centiday? Express your
100
answer as a decimal to the nearest tenth.
12. A centiday is
13. The temperatures rose from -3˚C at 7:00 am to 15˚C at 2:30 pm. Assume the
temperatures rose at a uniform rate.
a. What was the average rate of increase in temperature in degrees per hour
during that time?
b. What was the temperature at 12 o’clock?
c. At what time was the temperature 5˚C?
14. If Andy bikes at a steady rate of 15 miles per hour, how many miles will he bike in
four minutes?
15. What is the angular speed of the big hand on a clock, in degrees per minute?
16. Speeds at sea and in the air, are often measured in knots. The speed of one knot
means one nautical mile per hour. A nautical mile is a distance of approximately 1.85
km.
a. What is the speed in km/h of a 15 knot wind?
b. A ship steams 30 nautical miles at a speed of 12 knots, then a further 20
nautical miles at a speed of 15 knots. What is the total time taken?
c. How far, in kilometers, would a plane fly at a speed of 240 knots, between
10:30 am and 1:15 pm?
d. A boat sails 850 m in 6 minutes. What is the equivalent speed in knots?
Answer correct to 1 decimal place.
17. Julia jogs at a rate of six miles per hour. What is her rate in feet per second?
Express your answer as a decimal to the nearest tenth.
Note: There are 5280 ft in 1 mile.
18. If one liter equals approximately 1.06 quarts, and one quart contains 32 ounces, how
many ounces are in 3 liters? Express your answer as a decimal to the nearest
hundredth.
19. The courthouse in Pleasantville measures 1980 ft x 440 ft. One square mile equals
640 acres. How many acres does the courthouse cover?
20. Thanksgiving Travel. Zach is planning to drive to his parents’ house in Philadelphia
for Thanksgiving. He lives in Pittsburgh so he can either take Route 30, which is a
scenic drive, or take the Pennsylvania Turnpike. He’s trying to decide between the
two routes. Here are the details that he’s considering:
Pennsylvania Turnpike:
- Distance is 305 miles each way.
- Toll road charge is $9.00 each way.
- 1/61 of the trip is city driving and 60/61 is high way driving
Route 30:
- Distance is 372 miles each way.
- No tolls
- 1/3 of the trip is city driving, and 2/3 is highway driving
Gas costs on the average $1.60 per gallon. Zach’s car gets 25 miles per gallon in
the city and 30 miles per gallon on the high way. (Route 30 involves more city
driving because he’d be driving through a lot of small towns.)
Explain which plan is most economical (not considering the time involved or the
wear and tear on his car)
21. In march 2000, demolition experts used 4461 pounds of gelatin dynamite to bring
down the Kingdome, a stadium in Seattle. The dynamite was stuffed into 5905
holes throughout the dome. What was the mean number of ounces of dynamites per
hole? Round your answer to the nearest whole number.
Note: 1 pound = 16 ounces.
22. Given that t and w are in the ratio 5:6, and w and y are in the ratio 4:7, what is the
ratio t:w:y?
23. While driving down the Aegean coast you see a sign that says gasoline costs
229,704 Turkish Lira per Liter. The current exchange rate is 410,000 TL to a
United States dollar. How much does the gasoline cost in dollars per gallon?
(1 gal 3.784 L) Round to the nearest cent.
24. Using A,B,C,D and E, list the corresponding speeds in order from fastest to
slowest. Use 1 mile =5280ft, 1 km=0.62 miles and 1 m= 3.28 ft.
a. 10,000 ft/min
b. 60 miles /hr
c. 100ft/sec
d. 100km/hr
e. 30m/sec
Ratios and Unit Rates – Blue Solutions
Part 1. Problem Involving RATIOS
1. 3
2.
The ratio of 2 to
5
4 is 2
7
5
4
7
2 7
5 4
14
20
7
. As a decimal, 7 = 0.7
10
10
3. If we rewrite the ratio c : d = 1 : 5 as the equivalent ratio c : d = 6 : 30, we can include d in the ratio a
: b : c : d = 3 : 5 : 6 : 30 and the ratio b : d= 5 : 30, which can be rewritten as the common fraction
5/30 or 1/6
4. A person weighing 1/300 of one ton would weigh 2000 ÷ 300=6.7 pounds, to the nearest tenth.
5. We are told that the ratio of boys to girls in a class of 35 students is
3 : 4. This means that for every three boys in the class, there are
four girls. We could draw boys and girls, but to make this a little
easier, we will use common symbols to represent a boy (arrow) and a
girl (plus). Figure 1 shows our ratio of three boys to four girls. The
glitch, though, is that we have more than 7 students. We need 35
students. Let’s add on three more boys and four more girls. We still
have only 14 students, as shown in figure 2. Notice that to keep our
ratio correct, we must add on three boys and four girls each time,
which is seven students. It will take five groups of these seven
students to arrive at our class total of 35 students (figure 3). So
there are five groups of these three boys, which is a total of 5x3 =15
boys. There are also five groups of these four girls, which is a total
of 5x4 = 20 girls. There is then 20-15=5 more girls than boys in the
class. A shortcut would have been to see that each group of seven
students has one more girl than boy. Since there are five groups of
seven, there are five more girls than boys.
6. $64
Figure 1
Figure 2
Figure 3
7. Jodie gets $12.80 and Sarah gets $9.60
8.
a.
1000 mL orange, 400 mL lemon, 1600 mL mineral water
b.
1875 mL of drink
9. 0.75 m (or 75 cm)
What are we asked to find? The number of blue marbles in a jar.
10. Find Out
Choose a strategy
We can generate fractions equivalent to 1 . The numerator will represent
3
the number of red marbles, and the denominator will represent the
number of total marbles. Then, we can add 1 to the numerator and
denominator to simulate adding one red marble until we find a ration
equivalent to 3 .
8
A table showing equivalent fractions and the resulting fractions when the
numerator and denominator are increased by 1 is shown below.
Solve it
Fraction Equivalent to 1
3
Fraction when Numerator is Increased by 1
1
3
2
6
3
9
4
12
5
15
2
4
3
7
4
10
5
13
3
6
=
16 8
The ration will be 3 when there are initially 5 red marbles and 15 marbles
8
Look Back
total. Hence, there must be 10 blue marbles in the jar.
Does this answer check? Yes. There are 5 red marbles and 15 marbles
total, so the ratio is 1 . If a red marble were added, there would be 6 red
3
marbles and 16 marbles total, which is ratio of 3 .
8
Problems Involving and Conversions
1. It costs $45 per day to rent the car. That leaves 125-45 = $80 for mileage. At 35¢ per mile, John
could drive 80 = 228.57 miles. The greatest whole number of miles he could drive so that his total
0.35
is less than $125 is 228.
2. Jessica ran for 2 hours and 55 minutes, which is 120+55=175 minutes. If it took Jessica 175 minutes
to complete 7/8 of her run, then 1/8 of her run took 175 ÷ 7 = 25 minutes. (This assumes that all of
the 7 segments, each 1/8 of the run were run at the same pace.) Since she has 1/8 of the run to
complete, it will take her 25 minutes.
3. First convert all times to minutes.
Xa: 2 x 60 + 15= 135 minutes
Yo: 1 x 60 +40= 100 minutes
Zu: 2 x 60 +30 = 150 minutes
Together, they worked 385 minutes and were paid $77. The rate is 385 ÷ 7 = 20¢ per minute. Since
Xa worked 135 minutes, he should get 135 x 20 = 2700¢, or $27.
4. If the faucet leaks at a rate of 1 pint every 2 hours, then it leaks 12 pints in 1 day, and it would leak
48 pints in 4 days. There are 4 quarts in each gallon, and 2 pints in a each quart. Thus, there are 8
pints in one gallon. Consequently, 48 pints is 48 ÷ 8 = 6 gallons, so the faucet leaks 6 gallons of water
in 4 days.
5. In order to determine the car’s average speed, several pieces of information must be obtained. First,
the next possible number that reads the same forward and backward greater than 62,926 must be
found. Second, the number of miles traveled must be determined. Third, the amount of time it took to
travel that distance must be calculated. Finally, the average speed can be determined by dividing the
number of miles by the number of hours.
The first step is finding the next palindrome. The hundreds digit could be changed, and the resulting
number would still be a palindrome. However, changing the hundreds digit decreases the number of
miles and that can’t happen. Therefore, change the thousands digit. To keep symmetry, change the
tens digit. The least possible change is to change them from 2 to 3. Since we want the least possible
mileage, change the hundreds digit to 0. This gives the next possible mileage that reads the same
forward and backward as 63,036.
The car therefore travelled 63,036- 62,926 = 110 miles. It travelled this distance from 10:18 to
11:58, a time of 1 hour, 40 minutes or 1 2 hours. A car travelling 110 miles in 1 2 hours has an average
speed of 110 = 66 miles per hour.
12
3
3
6. Because each of the 7.60 Euros was the same as 155 yen, we simply need to multiply 7.60 Euros by 155
Japanese yen per euro to get 1178 yen.
7. There are 60 x 60 = 3600 seconds in an hour, so if the car traveled 90 feet per second for an hour, it
would go 90 x 3600= 324,000 feet. There are 5280 feet in one mile, so this is 324,000 ÷5280 = 61
mph, to the nearest whole number.
8. If Paitlyn baby-sits for 4.5 hours at the rate of $5.50 per hour, then she earns 4.5 x $5.50= $24.75
per week. It will take her $750 ÷ $24.75/week = 30.303 weeks or 31 weeks to earn enough money to
buy the computer.
9.
The rise to run for this step is
6
3
1
13
4
4
27 53
4 4
27
53
0.51 .
This is within the acceptable
limits.
10. If Howard ran the 13.1 miles in 2 hours, then he ran 13.1 miles in 120 minutes. The first 10 miles took
95 minutes. That leaves 13.1 – 10 = 3.1 miles to be run in 120 – 95 = 25 minutes. The average pace is 25
÷ 3.1 = 8.1 minutes per mile.
11. First of all, we know that it will take less than one hour for light to go 121,000,000 miles since it
ravels 670,000,000 miles in a full hour. Dividing 121 by 670, we get 0.1806 hours or 0.1806 x 60=
10,836 minutes or 10,836 x 60 650 seconds.
12. There are 60 minutes in an hour and 24 hours in a day, so there are 60 x 24 = 1440 minutes in a day. A
centiday is thus 1/100 x 1440 = 14.4 minutes.
13.
a.
2.4 deg/h
b.
9ºC
c.
10:20 am
14. One way to approach this problem is to see it as “Rate x Time = Distance” problem. We know the rate
is 15 mph, and he is biking for four minutes, which is 4 = 1 of an hour. (Remember that if our rate
60
15
is in miles per hour, then our time must be in hours, and our distance will be in miles.) Now we can
calculate that 15 mph x 1 hr = 1 mile.
15
We also could view this as a proportional reasoning problem. We know that he travels 15 miles in 60
minutes, and we want to know the number of miles he travels in four minutes. This leads to the
proportion 15 =
60
then gets us to
x
4
x=
. Setting the cross products equal, we have 60x = 4(15). Dividing both sides by 60
4 (15) 60
=
= 1 mile.
60
60
15. 6 deg/min
16.
a.
27.75 km/h
b.
3h 50 min
c. 1221 km
d. 4.6 knots
17. Let’s agree first that (1) multiplying an expression by 1 does not change the value of the expression
and (2) a ratio is equal to 1 if the value in the numerator is equal to the value in the denominator.
Keeping these two facts in mind, consider 6 mph or the ratio 6 miles . We want to change this to
1 hour
? feet . Take a look at the product
? sec
6miles 1hour 1min 5280 ft
1hour 60 min 60sec 1mile
. The first ratio is our 6
mph, and each of the next three ratios is equivalent to 1 since the value in the numerator of each
ratio is equal to the value in its denominator. This ensures that the expression is still equivalent to 6
mph. “ Cancelling” factors in the numerators with factors in the denominators is a common practice
when we are multiplying fractions, and after performing the same type of procedure with the units
in the expression above., we are left with
6 5280 ft
.
60 60sec
Dividing out factors of 6, 10 and another 6
from the numerator and denominator, we have 88 = 8.8 feet/second.
10
18. There are 1.06 (3) (32) = 101.76 ounces in 3 liters.
19. There are 5280 feet in one mile, so there are 5280 x 5280 square feet mile. Since there are 640
acres in a square mile, then there are (5280 x 5280) ÷640=43,560 square feet in one acre. The
courthouse is 1980 x 440 = 871,200 square feet. Consequently, it covers 871,200 ÷ 43,560=20 acres.
20. Thanksgiving Travel. The most economical plan is to go to Route 30. Plan: My plan is to figure out the
prices of gas plus toll of Pennsylvania Turnpike and Route 30.
Information that I have: Pennsylvania Turnpike – have a distance of 305 miles, 1/61 is city driving,
60/61 is highway driving, has a toll for $9.00. Route 30 – have a distance of 372 miles, 1/3 is city
driving, 2/3 is highway driving, has no toll, 25 miles of city driving = one gallon, 30 miles of highway
driving = one gallon and one gallon is $1.60.
Information that I need: Money needed for gas on city driving, money needed for gas on highway
driving, and money needed for two way reach [ ( D+T) x2, D for total money needed for the one way
driving, T for toll, and *2 for two way (coming and leaving).] on Pennsylvania Turnpike and Route 30.
What I did: Pennsylvania Turnpike- since the miles on city driving and highway driving are fractions
with the denominator of 61. So I divided the total miles (305) by the denominator (61) and multiplied
the numerator (1 or 60) and got 5 miles for the quotient and 5 miles for the city driving , 300 miles
for the highway driving for the product. We know that in city driving 25 miles = one gallon, so 5 miles
is 5/25 or 1/5 of a gallon. We also know that in highway driving 30 miles = 1 gallon, so 300 miles is
300/30 or 10 gallons. Finally the total gallon for all the driving using the Pennsylvania Turnpike is 10
1/5 gallons. Now, we know that a gallon is $1.60, so 10 1/5 is 10 x1.6 + 1/5 x1.6 or 16+0.32 or $16.32.
Route 30- If you follow the step of what I did in Pennsylvania Turnpike you would get: 124 miles for
city driving, 248 for highway driving, 4.96 gallons for city driving, 9.92 gallons for highway driving,
$21.17 for total money needed for the one way driving.
Solving Equations: Using the equation I created, we can determine the way, Pennsylvania Turnpike or
Route 30.
Pennsylvania Turnpike
(D+T) x 2
= (16.32x 9) x 2
=25.32x2
=$50.64
Route 30
(D+T) x 2
=( 21.17+0) x 2
= 21.17x2
=$42.34
We know that 42.34 is less than 50.64, so Route 30 is the most economic plan.
21. Multiplying the 4,461 pounds of gelatin dynamite by 16 ounces per pound gives us 71,376 ounces.
Dividing 71,376 ounces by 5905 holes, we get about 12 ounces per hole.
22. t:w:y = 10:12:21
23. about $2.12/ gal
24. To compare the various speeds, we can convert them all to the same units of measure. Let’s use feet
per second. Speed A is 10,000 feet per minute, which is 10,000 ÷ 60 166.7 feet per second. Speed B
is 60 miles per hour, which is (60 x 5280) ÷ (60 x 60) = 88 feet per second. Speed C is 100 feet per
second. Speed D is 100 kilometers per hour, which is (100 x 0.62 x5280) ÷ (60 x 60) 90.9 feet per
second. Finally, speed E is 30 meters per second, which is 30 x 3.28 = 98.4 feet per second. Listing
the speeds in order from fastest to slowest, we get ACEDB.
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources were not known.
Problems
Bibliography Information
20 (PART 2)
The Math Forum @ Drexel
(http://mathforum.org/)
1 – 5, 10 (PART 1)
1 – 12, 14 , 17 –
19, 21,22,24
(PART 2)
Math Counts (http://mathcounts.org)
6 – 9 (PART 1)
13, 15 – 16
(PART 2)
Thomas, Alice, and Margaret Jordan. Maths
in the Fast Lane. Putney: Phoenix Education,
1998. Print