**Comparing Rates :**

In this section, we are going to learn, how to compare the given two measures by converting them into unit rates.

Let us look at an example to understand "Comparing Rates"

**Example :**

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

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

Cost of 1 doll = 120 / 8 Cost of 1 doll = $ 15 |
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.

To have better understanding on comparing rates, let us look at 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. **

Earning in 8 hrs = $57.60 Earning in 1 hr = 57.60/ 8 Earning in 1 hr = $7.20 |
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. **

Distance in 3 hrs = 120 miles Distance in 1 hr = 120 / 3 Distance in 1 hr = 40 miles |
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. **

No.gallons in 2 days = 5.2 No.of gallons in 1 day = 5.2/2 No.of gallons in 1 day = 2.6 |
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 ⋅ 500 grams Price of 1 kg = 2 ⋅ 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. **

Distance in 2 min = 6 miles Distance in 1 min = 3 miles 1 hour = 60 minutes Distance in 1 hr = 60 ⋅ 3 Distance in 1 hr = 180 miles |
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 **

Speed = (2/5) / (1/7) Speed = (2/5) ⋅ (7/1) Speed = 14 / 5 Speed = 2.8 miles per hour |
Speed = (3/5) / (2/7) Speed = (3/5) ⋅ (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 % **

Percent = [240/300] ⋅ 100 % Percent = 80 % |
Percent = [328/400] ⋅ 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 % **

Percent = [250/5000] ⋅ 100% Percent of income = 5 % |
Percent = [280/7000] ⋅ 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 "Comparing rates", we hope that the students would have understood how to solve problems using unit rates.

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