COMPARING RATES

About "Comparing rates"

Comparing rates : 

"Comparing rates" is nothing but comparing 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

Cost of 1 doll  =  $ 15

6 dolls cost $102

Cost of 1 doll  =  102 / 6

Cost of 1 doll  =  $ 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.   

Comparing rates - Practice problems

To have better understanding on comparing unit rates, 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

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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 

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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

Let us look at the next problem on "Comparing rates"

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

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