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.

Leak in 15 min is = 2/3 gal Leak in 1 hour is = (2/3) x 4 gal = 8/3 gal = 2.67 gallons |
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

**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"

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 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"

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. **

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.

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. **

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.

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. **

No.gallons in 2 days = 5.2 No.of gallons in 1 day = 2.6 |
No. 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 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"

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. **

Distance (2 min) = 6 miles Distance (1 min) = 3 miles 1 hour = 60 minutes Distance (1 hr) = 60 x 3 Distance (1 hr) = 180 miles |
Distance (1.5 hrs) = 225 miles Distance (1 hr) = 225/1.5 Distance (1 hr) = 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 **

Speed = (2/5) / (1/7) Speed = (2/5) x (7/1) Speed = 14 / 5 Speed = 2.8 miles/hour |
Speed = (3/5) / (2/7) Speed = (3/5) x (7/2) Speed = 21 / 10 Speed = 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 % **

Percent = [240/300]x100 Percent = 80 |
Percent = [328/400]x100 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]x100 Percent of income = 5 |
Percent =[280/7000]x100 Percent of income = 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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