Mixture A: An herbalist is making a medicinal blend of herbs. The blend includes ground ginseng, which costs $36.20 per kilogram, and golden chia, which costs $19.60 per kilogram. Overall, the blend costs $26.99 per kilogram. If the herbalist already has 3 kilograms of ginseng prepared, how many kilograms of golden chia should she add? (Round to the nearest tenth)

Let x = kilograms of golden chia to add

$/kilogram / kilograms
Ginseng / 36.20 / 3
Golden chia / 19.60 / x
Blend / 26.99 / x +3

36.2(3) + 19.6x = 26.99(x + 3)

108.6 + 19.6x = 26.99x + 80.97

7.39 x = 27.63

x 3.7 kilograms

The herbalist must add 3.7 kilograms of golden chia to the blend.

Mixture B: How many pounds of dark chocolate worth $1.19 a pound must be mixed with 10 pounds of milk chocolate worth 93 cents a pound to produce a mixture worth $0.99 a pound.

d = pounds of dark chocolate to add

$/lb / lbs
Dark Chocolate / 1.19 / d
Milk Chocolate / 0.93 / 10
Mixture / .99 / d + 10

1.19d + 0.93(10) = 0.99(d + 10)

1.19d + 9.3 = 0.99d + 9.9

0.20 x = 0.6

d = 3 pounds

Three pounds of dark chocolate should be mixed.

% Mixture C: Alli invested a portion of $5,000 at 0.3 % interest and the balance at 0.25% interest. How much did she invest at each rate if her total income from both investments was $14.25?

Let x= the amount of money Alli put into the 3% account

% invested / $ / $ interest
0.3% / x / .003x
0.25% / 5000-x / .0025(5000-x)
- / 5000 / 14.25

0.003x + 0.0025(5000-x)= 14.25

0.003x + 12.5 - 0.0025x= 14.25

0.0005x = 1.75

x = $ 3500

5000 - x = $1500

Alli invested $3500 at 0.3% interest and $1500 at 0.25% interest.

% Mixture D: For a big cleaning job, Dee wants to mix some ammonia-based cleaners to get a certain concentration of ammonia. She has 9 gallons of cleaner that is 5% ammonia. She also has a large supply of cleaner that contains 15% ammonia. How many gallons of the 15%-ammonia cleaner does Dee need to add to the 5%-ammonia cleaner to obtain a batch that contains 6% ammonia?

Let x = gallons of 15% ammonia solution

% ammonia / gallons
5% / 9
15% / x
6% / x +9

.05(9)+ .15x = .06(x + 9)

.45 + .15x = .06x + .54

.09x = .09

x = 1 gallon

Dee must add 1 gallon of 15% ammonia cleaner.

Motion E: Two groups of friends are taking taxis to a restaurant in philly. The first taxi leaves from Old City and travels at 33 miles per hour. At the same time, the other taxi leaves from South Street and travels 43 miles per hour. The two taxis are 12 miles apart and heading directly towards each other. How many minutes will it take for the taxis meet? (Round to the nearest tenth)

Let t = the time it takes for the taxis to meet

rate (mph) / time (h)
1st taxi / 33 / t
2nd taxi / 43 / t

33t + 43t = 12

76t = 12

t = 12/76 hours

9.5 minutes

The taxis will meet after 9.5 minutes.

Motion F: Ms. Cantello’s class is on a ferry that is traveling 34 miles per hour. The students notice that they are passing right by a fishing boat that is traveling in exactly the opposite direction at 46 miles per hour. How many minutes will it takefor the two vessels to be 3 miles apart?

Let t = the time until the vessels are 3 miles part

rate (mph) / time (h)
Ferry / 34 / t
2nd taxi / 46 / t

34t + 46t = 3

80t = 3

t = 3/80 hours

= 2 ¼ minutes

It will take 2.25 minutes (2 minutes 15 seconds) for the vessels to be 3 miles apart

Motion G: A disabled sailboat is drifting on a current that is moving at 6 miles per hour. Fortunately, it's drifting towards a Coast Guard station 35 miles away, which picks up the distress call and sends a rescue craft directly towards the sailboat. The rescue craft can travel 45 miles per hour in still water. How long will it be before the two vessels meet? Round to the nearest minute.

Let t = time before the vessels meet

rate (mph) / time (h)
Sailboat / 6 / t
Coast Guard / 45 - 6= 39 / t

6t + 39t=35

45t =35

t = 35/45 hours

47 minutes

The rescue craft will meet the disabled sailboat after 47 minutes.

Rate H:An express train travels 150 km/h from Florence to Rome, where the local train travels at an average speed of 95 km/h and takes 53 minutes longer. How far apart are the Italian cities to the nearest kilometer?

Let t = the time for the express train to travel from Florence to Rome

rate (km/h) / time (h)
Express / 150 / t
Local / 95 / t +53/60

150t = 95(t + 53/60)

150t = 95t + 1007/12

55t = 1007/12

t = 1007/660

d= 150(1007/660)

229 km

Florence and Italy are 229 km apart.

Rate I:The secretary to the superintendent wants to send a letter to every parent to inform them about the upcoming Keystone Exams. At 8:30am, she starts making copies on the new copier that copies pages at a rate of 83 pages per minute. At 8:50am, she realizes she could continue the job on the old copier that copies at a rate of 62 pages per minute. If she needs to make 5400 copies, at what time will the copying job finish?

Let t = the time it takes for the secretary to make 5400 copies

Pages/min / min
New copier / 83 / t
Old copier / 62 / t-20

83t + 62(t - 20) = 5400

83t + 62t - 1240= 5400

145t = 6640

t 46 minutes

46 minutes after 8:30 = 9:16

The secretary will finish copying at 9:16 am.