COMPARING RATES

In this section, you will learn, how to compare the given two measures by converting them into unit rates.

Example :

Compare "8 dolls cost $120" and "6 dolls cost $102".

To compare the given measures, convert than in to unit rates.

8 dolls cost $120

Cost of 1 doll = 120/8

= $15

6 dolls cost $102

Cost of 1 doll = 102/6

= $17

" 8 dolls cost $120" is less than "6 dolls cost $102"

Because, unit rate in "8 dolls cost $120" is $15. But the unit rate in  "6 dolls cost $102" is $17.

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"

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

= $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"

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

= $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.

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 

= 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.

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

= 2.6

Rosy

No. gallons in 3 days :

= 7.5

No. of gallons in 1 day :

= 7.5/3

= 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(500) grams 

Price of 1 kg :

= 2(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"

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 1 hr :

= 60(3)

= 180 miles

Daniel

Distance in 1.5 hours :

= 225 miles 

Distance in 1 hour :

= 225/1.5 

= 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)

= (2/5)  (7/1)

= 14/5

= 2.8 miles per hour

Declan

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

= (3/5)  (7/2)

= 21/10

= 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 is

= [no. of correct answers/Total no. of questions] x 100% 

Daniel

Percent :

= [240/300] ⋅ 100%

= 80%

Deborah

Percent :

= [328/400]  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]  100%

 = 5%

Plan B 

Percent of income :

= [280/7000]  100%

= 4%

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

Hence, plan A is better.

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