Unit Rates & Fractions
A pack of 8 markers costs $11.60. What is the cost per marker?
$1.45 per marker.
11.60 ÷ 8 = 1.45.
A museum sells tickets at a constant price.
| Tickets, t | Cost, c (dollars) |
|---|---|
| 2 | 37 |
| 5 | 92.50 |
| 8 | 148 |
Write an equation relating c and t.
c = 18.50t.
37 ÷ 2 = 92.50 ÷ 5 = 148 ÷ 8 = 18.50 dollars per ticket.
A scale drawing uses 1 inch to represent 12.8 feet. A model boat is 6 inches long. What is the actual length of the boat?
76.8 feet.
12.8 × 6 = 76.8.
A bicycle regularly costs $180 and is on sale for $144. What percent of the regular price is saved?
20%.
Savings: 180 − 144 = $36.
36 ÷ 180 × 100 = 20%.
A salesperson earns $420 per week plus 6% commission on sales. Last week sales totaled $2,350. What were the total earnings?
$561.00.
Commission: 2,350 × 0.06 = $141.
Total: 420 + 141 = $561.
A printer uses 2/3 of a cartridge to print 250 pages. At this rate, how many pages can it print using one full cartridge?
375 pages.
250 -: (2/3) = 375
The table shows the cost of ribbon.
| Length (yards) | Cost (dollars) |
|---|---|
| 3 | 7.20 |
| 6 | 14.40 |
| 9 | 21.60 |
| 12 | 28.80 |
Is the relationship proportional? Explain and identify the unit rate.
Yes. Each cost divided by its length equals 2.40.
The unit rate is $2.40 per yard.
A map uses a scale of 3 centimeters to 14 kilometers. Two towns are 7.5 centimeters apart on the map. Write and solve a proportion to find the actual distance.
35 kilometers.
3/14 = 7.5/d
3d = 105, so d = 35.
A jacket costs $64 before a sale. Store A discounts it by 25%; Store B discounts it by 15%. How much less is the sale price at Store A?
$6.40 less.
Store A: 64 × 0.75 = $48.00.
Store B: 64 × 0.85 = $54.40.
54.40 − 48.00 = 6.40.
A restaurant meal costs $85 before tax and tip. Sales tax is 8%, and the family leaves an 18% tip based on the pretax meal cost. What is the total cost?
$107.10.
Tax: 85 × 0.08 = $6.80.
Tip: 85 × 0.18 = $15.30.
85 + 6.80 + 15.30 = $107.10.
Maya earns $34.50 for 2 7/8 hours of pet care. At the same rate, how much will she earn for 4 1/3 hours?
$52.00.
34.50 -: (2 7/8) = 12
$12 per hour.
12 xx (4 1/3) = 52
A delivery service charges these amounts:
| Distance (miles) | Cost (dollars) |
|---|---|
| 2 | 9 |
| 4 | 15 |
| 6 | 21 |
Leo says the relationship is proportional because the cost increases by $6 every 2 miles. Is Leo correct? Explain.
No. Equal increases alone do not prove proportionality.
9/2 = 4.50, 15/4 = 3.75, and 21/6 = 3.50 dollars per mile. The ratios are not constant.
A trail mix recipe uses 3/5 cup of raisins for every 1 2/5 cups of cereal. How many cups of raisins are needed for 4 2/3 cups of cereal? Show a proportion or equivalent ratios.
2 cups of raisins.
(3/5) -: (7/5) = 3/7
cup of raisins per cup of cereal.
(14/3) xx (3/7) = 2
After a 30% discount, a backpack costs $39.20 before tax. What was its original price? Show how you know.
$56.00.
70% of the original price remains.
39.20 ÷ 0.70 = 56.
A store discounts a $150 pair of headphones by 20%. Then 7.5% sales tax is charged on the discounted price. What is the total cost?
$129.00.
Discounted price: 150 × 0.80 = $120.
Tax: 120 × 0.075 = $9.
Total: 120 + 9 = $129.
Brand A juice costs $2.34 for 18 fluid ounces. Brand B costs $3.04 for 16 fluid ounces. What is the difference in cost per fluid ounce? Show how you know.
$0.06 per fluid ounce.
Brand A: 2.34 ÷ 18 = $0.13/oz.
Brand B: 3.04 ÷ 16 = $0.19/oz.
0.19 − 0.13 = 0.06.
A proportional relationship compares time x, in hours, and water pumped y, in gallons. Its graph passes through (0, 0), (2, 150), and (6, 450). Explain the meaning of (1, 75) and write an equation.
(1, 75) means 75 gallons are pumped in 1 hour.
The constant of proportionality is 75 gallons per hour.
y = 75x.
A rectangular garden is 4.5 cm long and 3.2 cm wide on a drawing. The scale is 1 cm to 2.5 m. Find the actual area of the garden.
90 square meters.
Actual length: 4.5 × 2.5 = 11.25 m.
Actual width: 3.2 × 2.5 = 8 m.
Area: 11.25 × 8 = 90 m².
A wildlife center had 2,400 visitors in April. Attendance increased by 125% in May, then decreased by 20% in June. How many visitors came in June?
4,320 visitors.
May: 2,400 × 2.25 = 5,400.
June: 5,400 × 0.80 = 4,320.
An account earns 4.5% simple interest per year. How long will it take a $600 deposit to earn $81 in interest?
3 years.
I = Prt
81 = 600 × 0.045 × t
81 = 27t, so t = 3.
A recipe uses 5/6 cup of oats and 1 1/4 cups of milk. To keep the same ratio, how many cups of milk are needed for 1 cup of oats?
1 1/2 cups of milk.
Divide milk by oats:
(5/4) -: (5/6) = (5/4) xx (6/5) = 3/2
A spacecraft travels at a constant rate. Its distance-time graph has time x (hours) on the horizontal axis and distance y (miles) on the vertical axis.
The straight line passes through (0, 0) and (3, 45,000).
How far does the spacecraft travel in 4.8 hours? Show your reasoning.
72,000 miles.
45,000 ÷ 3 = 15,000 miles per hour.
15,000 × 4.8 = 72,000.
Two maps show the same trail. Map A uses 1 inch to represent 3.6 miles; the trail is 5 5/6 inches long on Map A. Map B uses 1 inch to represent 2.4 miles. How long is the trail on Map B?
8 3/4 inches.
Actual distance:
(35/6) xx 3.6 = 21
21 miles. Map B length:
21 -: 2.4 = 8.75
8.75 inches.
A tablet regularly costs $320. It is discounted 25%, then an additional 15% is taken off the sale price. Find the final price before tax and the overall percent discount.
$204.00; overall discount 36.25%.
320 × 0.75 = $240.
240 × 0.85 = $204.
(320 − 204) ÷ 320 × 100 = 36.25%.
Jordan deposits $480 in an account paying 3.5% simple interest per year. No money is added or withdrawn for 18 months. Find the interest earned and the total account balance.
$25.20 interest; $505.20 total.
18 months = 1.5 years.
I = 480 × 0.035 × 1.5 = $25.20.
Balance = 480 + 25.20 = $505.20.