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.

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

Solved 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

= $0.40

Cost of 6 pencils :

= $2.70

Cost of 1 pencil :

= 2.7/6

= $0.45

We get the lowest price per pencil $0.40 in "10 pencils cost $4"

So, "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

= $1.90

2 liters of milk at $2.70

Cost of 1 liter :

= 2.7/1.5

= $1.8

From the above unit rates, we get the lower price per liter of milk $1.8 in "2 liters cost $2.70"

So, "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

= $7.20

John

Earning in 12 hrs :

= $90

Earning in 1 hr :

= 90/12

= $7.50

From the above unit rates, John earns more than David per hour. 

So, 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

= 40 miles

Jose

Distance in 2 hrs :

= 84 miles

Distance in 1 hr :

= 84/2

= 42 miles

From the above unit rates, Jose covers more miles than David per hour. 

So, 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. of gallons in 2 days :

= 5.2

No. of gallons in 1 day :

= 2.6

Rosy

No. of gallons in 3 days :

= 7.5

No. of gallons in 1 day :

= 2.5

From the above unit rates, Lily prepares more gallons than day. 

So, 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 kg = 2 x 500 grams 

Price of 1 kg :

= 2 x 3.25

= $6.5

1.5 kilograms -----> $9.9

Price of 1 kg :

= 9.9/1.5

= $6.6

From the above unit rates, we get the lower price per kilogram $6.5 in "500 grams cheese cost $ 3.25"

So, "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 (2 min) :

= 6 miles

Distance (1 min) :

= 3 miles

1 hour = 60 minutes

Distance (1 hr) :

= 60 x 3

= 180 miles

Daniel

Distance (1.5 hrs) :

= 225 miles

Distance (1 hr) :

= 225/1.5

= 150 miles

From the above unit rates, Lenin covers more miles than Daniel per hour.

So, 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)

= (2/5) x (7/1)

= 14/5

= 2.8 miles/hour

Declan

Speed = (3/5)/(2/7)

= (3/5) x (7/2)

= 21/10

= 2.1 miles/hour

From the above unit rates, Shanel walk more miles than Declan per hour.

So, 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%

= 80%

Deborah

Percent :

= [328/400] x 100%

= 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 of income :

= [250/5000] x 100%

= 5%

Plan B

Percent of income :

= [280/7000] x 100%

= 4%

From the above percentages, plan A gives more income than plan B per $100 investment.  

So, plan A is better.

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