Ratios
Unit Rates
Writing and Solving Proportions
Percents
Percent Change
Proportional Relationships
100
At the local stationary store, there are 2 pens to 3 pencils. Write this as a proportion
2:3, 2 to 3, or 2/3
100
What is the definition of a unit rate?
A rate with a denominator of 1.
100
How do we check if a relationship is a true proportion?
We check that both cross products are equal to each other.
100
What is the definition of percent?
"per 100", or a fraction with a denominator of 100
100
How do we determine if we have percent increase or decrease?
If our change was positive (i.e. the new amount is larger than the original amount), we have percent increase.
If our change was negative (i.e. the new amount is smaller than the original amount), we have percent decrease.
100
What is the formula for a proportional relationship?
y = kx, where k is our constant of proportionality
200
Fully simplify the ratio 45:105
3:7
200
What is the (simplified) rate of a motorcycle that just traveled 14 miles in 12 minutes?
(14 mi)/(12min) = (7 mi)/(6 min)
200
solve for x:
x/4 = 7/2
2x = (7)(4)
2x = 28
x = 14
200
What is the proportion we use to represent percent problems?
(part)/(whol e) = %/100
200
What is the proportion we use to solve percent change problems?
(chang e)/(o rigi nal)=(% chang e)/100
200
How do we find the value of our constant of proportionality (k) in a proportional relationship?
Take any point from the relationship. The value of k is equal to the y value divided by the x value:
k = y/x
300
In a barnyard, the ratio of pigs to horses is 6 to 4. Write a simplified fraction to represent this ratio.
3/2
300
If a jogger runs 84 laps in 28 minutes, what is the rate at which this jogger runs?
84/28 = 3 laps per minute
300
Solve for a:
21/a = 63/45
63a = (45)(21)
63a = 945
a = 945/63
a = 15
300
What percentage commission does a salesperson make if they earn $151.34 from selling $658 worth of goods?
151.34/658 = x/100
15134 = 658x
x = 15134/658
x = 23%
300
Ten years ago, the price of gas was $2.49 per gallon, and today it is $3.55 per gallon. By what percent has the price of gas increased?
(3.55-2.49)/2.49 = x/100
1.06/2.49 = x/100
2.49x = 106
x = 42.57%
300
The table below represents the relationship between the price of pens, x, and the number of pens, y, at a stationery store. Write an equation to represent the relationship.
y=kx, k=y/x
k = 7.44/6 = 1.24
y = 1.24x
400
What is the missing number in the following ratio table representing the proportional relationship between distance and time?
The ratio between the two columns is 1:4.66.
so, we multiply 63 by 4.66 to get 294
400
If a 6 oz box of gummi bears costs $3.78, what is the unit price per ounce?
($3.78)/(6 oz) = ($0.63)/(oz)
400
A car travels at a rate of 357 miles every 4.5 hours. How many hours will it take this car to travel 952 miles?
357/4.5 = 952/x
357x = (952)(4.5)
357x = 4284
x = 12
400
A promotor for a gaming company makes a 28% commission for every game she successfully sells. If each game costs $55, how much money will she make selling 20 games?
First we will find how much she makes from selling 1 game, and then multiply that amount by 20.
x/55 = 28/100
100x = (55)(28)
100x = 1540
x = 1540/100 = $15.40
Now we multiply this amount by 20 games to get the total money made: $15.40 cdot 20 = $308
400
A basketball team had a 60% decrease in the number of victories it had this season compared to last season. If the team had 48 fewer victories, how many victories did they have this season?
48/x = 60/100
(48)(100) = 60x
4800 = 60x
x = 80 victories
400
The graph below represents the proportional relationship between the distance a pickup truck can travel in relation to time:
5.5 hours
500
What is the missing number in the following ratio table representing the proportional relationship between the weight of a space shuttle and the gallons of fuel needed to launch it into outer space?
The ratio between the two columns is 1:22.84.
So we divide 10,050 by 22.84 to get 440
500
If an office assistant can type 357 words in 6 minutes, what is the unit rate per hour that this office assistant can type?
356/6 = 59.5 words per minute
50.857 * 60 mi n utes = 3570 words per hour
500
Charcoal must be purchased for an upcoming bar-b-que. The store sells charcoal at a rate of $12.00 for every 2 lbs. How much will it cost for 40 ounces of charcoal?
($12)/(2lb) = ($6)/(1lb) = ($6)/(16oz) = ($.375)/(1oz)
.375/1 = x/40
1x=(.375)(40)
x = 15
500
What is the total cost for a $2,100 computer in New York if the state sales tax is 8.5%?
First we find the 8.5% tax, then we add it to the original cost.
x/2100 = 8.5/100
100x = (2100)(8.5)
100x = 17850
x = 178.50
Now we add this cost to the original cost of the laptop: 2100+178.5 = $2278.50
500
In 1997, the population of beavers in Beaverton, Oregon, was 345,960. If there are 40% more beavers in Beaverton today, how many beavers are there?
(chang e)/345960 = 40/100
(chang e)(100) = (345960)(40)+
chang e = 13838400/100 = 138,384
So we started with 345,960 and we increased by 138,384: 345,960 + 138,384 = 484,344
500
What is the constant of proportionality based on the graph below?
3/4