Unit rates with fractions :
We can use unit rates to simplify rates and ratios that appear complicated, such as those containing fractions in both the numerator and denominator.
To have better understanding on "Unit rates with fractions", let us look at the example given below.
Example :
Two pools are leaking. After 15 minutes, pool A has leaked 2/3 gallon. After 20 minutes, pool B has leaked 3/4 gallon. Which pool is leaking faster ?
To find which pool is leaking faster, we have to find leak per hour for each pool.
Pool A Leak in 15 min is = 2/3 gal Leak in 1 hour is = (2/3) x 4 gal = 8/3 gal = 2.67 gallons |
Pool B Leak in 20 min is = 3/4 gal Leak in 1 hour is = (3/4) x 3 gal = 9/4 gal = 2.25 gallons |
Compare the unit rates.
2.67 > 2.25
So, Pool A is leaking greater
To have better understanding on unit rates with fractions, let us look some practice problems.
Problem 1 :
Which is best,
10 pencils cost $4
or
6 pencils cost $2.70 ?
Solution :
To compare the given measures, convert them in to unit rates.
Cost of 10 pencils = $4 Cost of 1 pencil = 4 / 10 Cost of 1 pencil = $0.40 |
Cost of 6 pencils = $2.70 Cost of 1 pencil = 2.7 / 6 Cost of 1 pencil = $0.45 |
We get the lowest price per pencil $0.40 in "10 pencils cost $4"
Hence, "10 pencils cost $4" is the best deal.
Problem 2 :
Which is best,
2 liters of milk at $3.80
or
1.5 liters of milk at $2.70 ?
Solution :
To compare the given measures, convert them in to unit rates.
2 liters of milk at $3.80 Cost of 1 liter = 3.8 / 2 Cost of 1 liter = $ 1.90 |
2 liters of milk at $2.70 Cost of 1 liter = 2.7 / 1.5 Cost of 1 liter = $ 1.8 |
From the above unit rates, we get the lower price per liter of milk $1.8 in "2 liters cost $2.70"
Hence, "2 liters cost $2.70" is the best deal.
Problem 3 :
Who is better in earning,
David earns $57.60 in 8 hours
or
John earns $90 in 12 hours ?
Solution :
To compare the given measures, convert them in to unit rates.
David Earning in 8 hrs = $57.60 Earning in 1 hr = 57.60 / 8 Earning in 1 hr = $7.20 |
John Earning in 12 hrs = $90 Earning in 1 hr = 90 / 12 Earning in 1 hr = $7.50 |
From the above unit rates, John earns more than David per hour.
Hence, John is earning better.
Problem 4 :
Who is driving faster,
Alex covers 120 miles in 3 hours
or
Jose covers 84 miles in 2 hours ?
Solution :
To compare the given measures, convert them in to unit rates.
Alex Distance in 3 hrs = 120 miles Distance in 1 hr = 120 / 3 Distance in 1 hr = 40 miles |
Jose Distance in 2 hrs = 84 miles Distance in 1 hr = 84 / 2 Distance in 1 hr = 42 miles |
From the above unit rates, Jose covers more miles than David per hour.
Hence, Jose is driving faster.
Problem 5 :
Who is better,
Lily can prepare 10.4 gallons of juice in 4 days
or
Rosy can prepare 7.5 gallons of juice in 3 days ?
Solution :
To compare the given measures, convert them in to unit rates.
Lily No.gallons in 2 days = 5.2 No.of gallons in 1 day = 5.2/2 No.of gallons in 1 day = 2.6 |
Rosy No. gallons in 3 days = 7.5 No. of gallons in 1 day = 7.5/3 No.of gallons in 1 day = 2.5 |
From the above unit rates, Lily prepares more gallons than day.
Hence, Lily is better.
Problem 6 :
Which is best,
500 grams cheese cost $ 3.25
or
1.5 kilograms cheese cost $ 9.9
Solution :
To compare the given measures, convert them in to unit rates in price per kilogram.
500 grams -----> $3.25 1 kilogram = 2 x 500 grams Price of 1 kg = 2 x 3.25 Price of 1 kg = $6.5 |
1.5 kilograms -----> $9.9 Price of 1 kg = 9.9 / 1.5 Price of 1 kg = $6.6 |
From the above unit rates, we get the lower price per kilogram $6.5 in "500 grams cheese cost $ 3.25"
Hence, "500 grams cheese cost $ 3.25" is the best deal.
Problem 7 :
Who is driving faster,
Lenin covers 6 miles in 2 minutes
or
Daniel covers 225 miles in 1.5 hours ?
Solution :
To compare the given measures, convert them in to unit rates in distance per hour.
Lenin Distance in 2 min = 6 miles Distance in 1 min = 3 miles 1 hour = 60 minutes Distance in 1hr = 60x3 Distance in 1 hr = 180 miles |
Daniel Distance in 1.5 hrs =225 miles Distance in 1 hr = 225 / 1.5 Distance in 1 hr = 150 miles |
From the above unit rates, Lenin covers more miles than Daniel per hour.
Hence, Lenin is driving faster.
Problem 8 :
Who is better in walking,
Shanel walks 2/ 5 of a mile every 1/7 hour.
or
Declan walks 3/5 of a mile every 2/7 hour ?
Solution :
To compare the given measures, convert them in to unit rates in miles per hour (speed).
Speed = Distance / Time
Shanel Speed = (2/5) / (1/7) Speed = (2/5) x (7/1) Speed = 14 / 5 Speed = 2.8 miles per hour |
Declan Speed = (3/5) / (2/7) Speed = (3/5) x (7/2) Speed = 21 / 10 Speed = 2.1 miles per hour |
From the above unit rates, Shanel walk more miles than Declan per hour.
Hence, Shanel is better in walking.
Problem 9 :
Who is better,
Daniel answered 240 answers correctly out of 300 questions
or
Deborah answered 328 questions correctly out of 400 questions ?
Solution :
To compare the given measures, convert them in to percentages .
Percent= [no. of correct answers / To tal no. of questions] x 100 %
Daniel Percent = [240 / 300] x 100 % Percent = 80 % |
Deborah Percent = [328 / 400] x 100 % Percent = 82 % |
From the above percentages, Deborah answered more questions correctly than Daniel per 100.
Hence, Deborah is better.
Problem 10 :
Which is best,
Plan A : Income of $250 on $5000 investment
or
Plan B : Income of $280 on $7000 investment
Solution :
To compare the given measures, convert them in to percentages of income .
Percent of income = [Income / Investment] x 100 %
Plan A Percent = [250/5000] x 100% Percent of income = 5 % |
Plan B Percent = [280/7000] x 100% Percent of income = 4 % |
From the above percentages, plan A gives more income than plan B per $100 investment.
Hence, plan A is better.
After having gone through the stuff given on "Unit rates with fractions", we hope that the students would have understood how to solve problems using unit rates with fractions .
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