# 1. It Took 8 Hours to Drive Your Car 400 Miles. on Average, How Fast Were You Driving?

Car Problems

1. It took 8 hours to drive your car 400 miles. On average, how fast were you driving?

2. You need to drive to a job interview 250 miles away. It is currently 8am and the interview is at 1pm. If you can drive at an average speed of 55 mi/hr, will you get there in time?

If not, how many minutes late are you? If so, how many minutes early?

3. The distance between Cambridge and Wellesley is 10 miles. A person walks part of the way at 5 miles per hour, then jogs the rest of the way at 8 mph. Find a formula that expresses the total amount of time for the trip, T(d), as a function of d, the distance walked.

4. Granny lives 350 miles away, and you plan to drive to her house for Thanksgiving dinner, which starts at 3pm. You get an early start, leaving at 7am. Unfortunately, everyone else has the same idea, so there is lots of traffic. In fact, you are only able to manage an average speed of 35 mi/hr for the first half of this trip (175 miles). How fast do you need to drive for the rest of the trip if you want to get there in time for dinner?

5. You are on your way back to school after visiting your family for Thanksgiving. Unfortunately, you forgot to pack your textbooks (you were studying over the weekend). Luckily for you, your mom sees your books and decides to bring them to you. Suppose you left at 9am and are driving at a steady speed of 50 mph. If your mom leaves at 10:30am and drives at 70 mph, when will she catch up to you?

6. Larry lives in Los Angeles and Sam lives in Sacramento. They drive toward each other along the same route. Larry leaves at 9am and drives at a steady 60 mi/hr. Sam leaves at 10 am and drives at a constant 65 mi/hr. The distance between Los Angeles and Sacramento is 400 miles. Where are they and what time is it when they pass each other?

Mixture Problems

1. Initially there were 3 liters of blue paint and 8 liters of crimson paint. A paint job uses 20% of the blue paint and 80% of the crimson paint. What percentage of the total combined amount of paint is used during the job?

2. If beaker A contains a 15% solution of sulfuric acid and beaker B contains a 20% solution of sulfuric acid, how much from each beaker should be used to make 6 liters of a 18% solution of sulfuric acid?

3. I have two cans of paint. Can A contains 9 parts of yellow paint to 11 parts of blue paint. Can B is 80% yellow paint and the rest is blue paint. How much paint should I use from each can to obtain 1 liter of paint which is half blue and half yellow?

4. Purple-purple is made from equal parts blue paint and red paint. Ultra-purple is made from 7 parts blue paint and 3 parts red paint. Blue-purple is made from 8 parts blue paint and 2 parts red paint. How much Blue-purple and Purple-purple is needed to make each gallon of Ultra-purple?

5. Can A contains 2 liters of 30% oil and the rest vinegar, can B contains 2 liters 40% vinegar and the rest oil. Can C contains 3 liters of 80% oil and 20% vinegar. If half of Can B is added to Can A, then all of Can C is added to Can A, how much oil would be in Can A?

6. I have some cans of paint. Can A contains 10% blue and 90% red. How many liters of Can A should I mix with k liters of blue paint to get paint which is 60% blue?

ANSWERS:

Car Problems:

1) 50 mi/hr

2) Early by 27 minutes.

3) T(d) = d/5 + (10-d)/8

4) 58.3 mi/hr

5) 2:15 pm

6) 12:43 pm

Mixture problems:

1) 64%

2) 2.4 L from A, 3.6 L from B

3) 6/7 L from A, 1/7 L from B

4) 2/3 gallon of Blue-Purple, 1/3 gallon of Purple-Purple

5) 3.6 L

6) 4k/5 L

Solutions:

4. Purple-purple is made from equal parts blue paint and red paint. Ultra-purple is made from 7 parts blue paint and 3 parts red paint. Blue-purple is made from 8 parts blue paint and 2 parts red paint. How much Blue-purple and Purple-purple is needed to make each gallon of Ultra-purple?

First, notice that we can find the percentages for each type of paint:

Purple-Purple (P) has 50% blue and 50% red

Ultra-Purple (U) has 70% blue and 30% red

Blue-Purple (B) has 80% blue and 20% red

Now we can write 3 equations: one for the total amount of paint, and one for just the blue paint, or just the red paint. We really only need any 2 of these to solve the problem.

TOTAL:P + B = 1 the total is 1 gallon

BLUE:(50%)P + (80%)B = (70%)(1) remember – ultra-purple is 70% blue

RED:(50%)P + (20%)B = (30%)(1)

Now pick 2 equations and solve for the 2 unknowns:

## P = 1 – B

Plug this into the blue equation (we can drop the % signs):

(50)(1-B) + (80)(B) = (70)(1)

50-50B+80B=70 30B=20 B=2/3 gallon P=1/3 gallon

So we need 1/3 gallon of Purple-Purple and 2/3 gallon of Blue-Purple

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6. I have some cans of paint. Can A contains 10% blue and 90% red. How many liters of Can A should I mix with k liters of blue paint to get paint which is 60% blue?

Blue paint is 100% blue, so we can use that for “can B”.

Our formula for the blue paint should look like this:

(10%)(A) + (100%)(k) = (60%)(A + k)

we can drop the % signs, or use decimals if you prefer

10A + 100k = 60(A + k)or0.1A + k = 0.6(A + k)

now multiply it out and solve:

10A + 100k = 60A + 60kor0.1A + k = 0.6A + 0.6k

40k = 50Aor0.4k = 0.5A

4k/5 = Aor4k/5 = A

So we need to use 4k/5 liters from can A.

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