Word problems

VICs picking numbers procedures. (Only with variables in the answer choices)

Stage 1- after low hanging fruit 1. Go after low hanging fruit. 2. If there is only one variable in the answer choices and that variable is a percent, then it may be easy to figure out the prompt answer when this variable is either 0 or 100. 3. Do not use 2 if there is more than one variable. Stage 2- after low hanging fruit 1. Avoid 0 or 1 2. pick different numbers for different variables. The number should be different from those given in the problem 3. Don't pick numbers that are multiples of each other; picking different prime numbers is good. 4. Keep the numbers small.

Multiple workers/machines

The combined work rate of people/machines working together is the sum of the individual work rates

***Frank and Georgia started traveling from A to B at the same time. Georgia's constant speed was 1.5 times Frank's constant speed. When Georgia arrived at B, she turned around immediately and returned by the same route. She crossed paths with Frank, who was coming toward B, when they were 60 miles away from B. How far away are A and B?

a. Frank: D-60 = RT b. Georgia: D+60 =1.5RT c. Combined: D + 60 = 1.5(D -60) d. D + 60 = 1.5D - 90 e. D + 150 = 1.5D f. 150 = 0.5D g. D = 150/0.5 = 1500/5 = 300 (answer)

**Frank and Georgia started traveling from A to B at the same time. Georgia's constant speed was 1.5 times Frank's constant speed. When Georgia arrived at B, she turned around immediately and returned by the same route. She crossed paths with Frank, who was coming toward B, when they were 60 miles away from B. How far away are A and B?

a. Frank: D-60 = RT b. Georgia: D+60 =1.5RT c. Combined: D + 60 = 1.5(D -60) d. D + 60 = 1.5D - 90 e. D + 150 = 1.5D f. 150 = 0.5D g. D = 150/0.5 = 1500/5 = 300 (answer)

**Backsolving practice: IN a certain state, schools by 2% tax on food and 8% on stationery. A school placed a combined order of $500 on food and stationery, and paid 19 on tax on the order. How much of that money was spend on food A: 120 B: 250 C: 300 D: 350 E: 400

a. start with C: Food = 300; stationery =200 b. food tax = 300(0.02) = 6 c. stationery tax = 200(0.08) = 16 d. Total tax = 22 (too much). We can reason food needs to cost more and stationery needs to cost less. e. answer choice D: Food = 350; Stationery = 150 f. Food tax 350(0.02)= 7 g. Stationery tax= 150(0.08) = 12 h. 7 + 12 = 19 (answer choice D is correct)

**Most popular mixture question type: Suppose we start with unlimited supplies of 20% H solution and of 50% H solution. We combine X liters of the first with Y liters of the second to produce 7 liters of 40% H solution. What does X equal?

a. x + y = 7 b. Total solute: 7(0.4)= 2.8 liters c. 0.2X + 0.5Y= 2.8 liters d. go back to a (y= 7-x) e. 0.2X + 0.5(7-x)= 2.8 f. 0.2x + 3.5 - 0.5x = 2.8 g. times by 10 to get rid of decimals: 2x + 35 -5x = 28 h. 35-3x = 28 i. -3x = -7 j x= 7/3 liters (answer)

Work problem categories

a. Proportions b. multiple machines/multiple worker to see how much they get done together.

Practice with VICS: At the store, Sam bought a shirt and a toaster. There was an 8% sales tax on each item, and with tax, Sam paid a total of Q. If the price of the toaster before tax was T, what, in terms of Q and T, is the price of the shirt. A: 0.92(Q-T) B: 0.92Q-T C: 0.92(Q-1.08T) D: (Q-T)/1.08 E: (Q/1.08) - T

a. Q = 1.08(S+T) b. Q/1.08 = S+T c. (Q/1.08) -T = S (answer choice E) NOTE: VICS is a difficult concept. I may need to review the videos.

A lichen advances 4 cm each year across a rock slab. If this rate remains constant over time, how many years will it take to cross 30 meters? (1 m = 100 cm)

a. R = 4 cm/yr b. D = 30 m = 3000 cm c. 3000 cm = 4cm/yr (T) d. T = 3000 cm/ (4cm/yr) e. T = 750 yrs.

**A car moving at 72 km/hr moves how many meters in one second? (1 km = 1000 m)

a. R = D/T b. 1hr = 60 x 60 = 3600 s c. 72 km/hr = 72,000 m/3600 s = 20 m/s

Right now, Steve's age is half of Tom's age. In eight years, twice Tom's age will be five more than three times Steve's age. How old is Tom right now?

a. S = (1/2)T b. T = 2S (this avoids fractions) c. 2(T+8) = 3(S+8) + 5 d. 2(2S+8) = 3(S+8) + 5 e. 4S + 16 = 3S + 29 f. S + 16 = 29 g. S = 13 (plug back in to b) h. T = 26

**A sequence is defined by S sub n = (S(sub n -1) -1)(S (sub n-2) for n >2, and it has the starting values of S sub 1 = 2 and S sub 2 = 3. Find the value of S sub 6.

a. S sub 1 = 1 b. S sub 2 = 3 c. S sub 3 = (2-1)(2)= 4 d. S sub 4 = (4-1)(3)= 9 e. S sub 5 = (9-1)(4)= 32 f. S sub 6 = (32-1)(9) = 279

Picking numbers practice: Before January, the price of a dress was D and the price of a matching pair of shoes was H. In January, the price of the dress increased by 40% and the price of the shoes increased by 50%, and didn't change again in the following months. In March, Roberta bought both items with a 30% discount. If D = 5H, which of the following represents the amount that Roberta paid? A. D+40 B. D+40 -1 C. D + 2H D. 5.95H E. 1.21D

a. Suppose D = 200 and H = 40 b. After increases D = 280 and H= 60 c. 280 + 60 = 340 d. 30% decrease of 340 is 238 e. plugging the amounts for D and H before increase into each potential answer should yield 238 for the correct answer only. f. D is the only one that works (6-0.05)(40) = 240 - 2 = 238 (answer) Note: VICs is still very difficult for me. This will take some practice.

Translating word problems

1. 'is' or 'are' correspond to equals sign. 2. '50 more than B' corresponds to B+50 3. '50 less than B' corresponds to B-50

To detail a car means to clean it inside and out. Amelia can detail one car in four hours. When Amelia and Brad detail a car together, 1 car takes 3 hours. How long does it take Brad, working alone, to detail a car?

1. A + B = 1/3 2. A= 1/4 3. 1/4 + B = 1/3 4. B = 1/3 - 1/4 = 4/12 - 3/12 = 1/12 (Brad is one car every 12 hours).

Work Problem formula

1. A = RT 2. A = amount of work done 3. R = Work rate 4. T = time.

Arithmetic sequence

1. A sequence in which we add the same constant to get from each term to the next. (5, 12, 19...) the constant is 7. 2. any evenly spaced list is an arithmetic sequence. consecutive multiples of a number and consecutive odds and evens are all arithmetic sequences. Numbers which, when divided by the same divisor have the same remainder are also arithmetic sequences. the remainder is the a sub 1 term. 3. the nth term of an arithmetic sequence with an initial term a sub 1 and a common difference d is: a sub n = a sub 1 + (n-1)xd

Assigning variables

1. Choose a meaningful variable. A letter that begins the name of a person, for example, is helpful. 2. It could be helpful to assign a variable to the smallest value 3. It could be helpful to assign a variable to the target value. 4. If all the variables are related to one quantity, choose that quantity as the variable.

Age word problems (strategies)

1. It is usually easier to pick the variable to represent the age right now. (F = Frieda's age right now) 2. Use addition and subtraction to create expressions for ages at other times (e.g. F-5 = Frieda's age five years ago; F+7 = Frieda's age in 7 years.

Sums of sequences

1. The sum of any evenly spaced list that has N items (N is the number of items on the list) has the following formula: [N(a sub 1 + a sub last number in sequence)]/2

Average speed problems (to solve)

1. There is a D = RT for the first leg, there is a D = RT for the second leg, and D = RT for the trip as a whole. 2. average velocity = total distance/total time. 3. you need to find the time for both legs. before you can figure out the average velocity.

Backsolving definition and procedure

1. Use when word problems are mc with numerical answers. Procedure: a. assume one of the answers is correct. b. If the answer is incorrect, we can usually tell if the answer should have been bigger or smaller. c. Always begin with answer choice C

Growth and decay questions

1. Very rare 2. Take calculations at 1 change interval at a time.

Two travelers moving in the same direction (gaps)

1. We subtract the speed--bigger minus smaller Note: If faster traveler is in front, we are measure the expansion. I the slower traveler is in front we are measure the shrinkage. Note 2: set up a D = RT for the gap to save time.

Mixture Questions

1. concentration = (amount solute/total amount of solution) x 100 2. Solution: water + solute 3. when amounts of two different solutions are unknown, we have to set up simultaneous equations (most popular question type) 3.1 One equation will always be a "total" equation (mass, volume, weight, etc.). the other equation will be about the amount of solute.

Sequence

1. ordered list of numbers 2 Infinite, following a pattern. 3. the sequence as a whole is represented by a letter and an individual letter by a numerical subscript (e.g. a sub 5 = 28) 4. an algebraic formula is used to represent the repeating pattern 5. a sub n = n is the sequence of all positive integers. 6. a sub n = 2n -1 is the sequence of all positive odd numbers. 7 a sub n = 2n is the sequence of all positive even numbers. 8. a sub n = 7n is the sequence of all positive multiples of 7 (this works for any factor times n) 9. a sub n = n^2 is the sequence of all perfect squares. 10 a sub n = n^3 is the sequence of all powers of 3.

Two travelers moving in opposite directions (gaps)

1. we add the speeds of two travelers moving in opposite directions (moving away from or toward each other). Toward: The sum is the speed at which the gap is shrinking. Away from: The speed at which the gap is expanding. Note: set up a D = RT for the gap to save time.

Translation practice: Twice A is 100 less than three times B

2A = 3B - 100

When isotope QXW radioactively decays, it loses exactly half of its mass in each three-day period. Suppose scientists start with a 96 gram sample of pure isotope QXW on a certain day. What will be the remaining mass in 12 days.

Day 0: 96 Day 3: 48 Day 6: 24 Day 9: 12 Day 12: 6 grams (answer).

**What is the sum of al the multiples of 20 from 160 to 840 inclusive?

Determine the number of terms first a. 160 = 8(20), the 8th multiple of 20 b. 840 = 42(20), the 42nd multiple of 20 c. Inclusive counting: 42 - 8 + 1 = 35 d. number of pairs = 17.5 e. Sum of list: 17.5(160+840) = 17.5(1000)= 17,500 (answer)

Motion questions

Distance Rate: rate or speed Time b. D = RT c. Units of rate determines units for distance and time.

Suppose we start with 8 liters of 60% H solution. We add 4 liters of C% H solution, and the result is 12 liters of 50% H solution. What is C?

a. solute in first solution: 8(0.6)= 4.8 liters b. solute in result: 12(0.5)= 6 liters c. Thus, solute added = 6-4.8 = 1.2 liters d. C = 1.2/4 = .3 = 30% (answer)

Inclusive counting

Adding 1 to the answer when the start value and ending value are included in what we are counting.

Sam's car was fined for parking when he gave Joe and Peter a ride, so they decided to help Sam pay the fine. Joe paid 3 more than 1/4 of the fine and Peter paid 3 less than 1/3 of the fine, leaving Sam 4 less than 1/2 the fine to complete the payment. How much did Sam pay?

Joe: F/4 + 3 Peter: F/3 -3 Sam: F/2 -4 a. (f/4) + 3 + (f/3) -3 + (f/2) -4 = f b. (f/4) + (f/3) + (f/2) -4 = f c. 3f + 4f + 6f -48 = 12f d. 13f - 48 = 12f e. f - 48 = 0 f. f = 48 g. Sam = (48/2) - 4 h. Sam = $20 (answer)

A machine working at a constant rate, makes 36 staplers in 28 minutes. How may staplers does it make in 1hr 45 min.

Proportion method quickest a. 36/28 = S/105 b. 9/7 = S/105 c. 9/1 = S/15 d. S = 9 x 15 = 135 (answer)

In a certain company, 50% of Ed's salary is 20% of Ruth's salary. If the difference between their salaries is 90,000, what is Ruth's salary?

a. (1/2)E = (1/5)R b. R-E = 90,000 c. E = (2/5)R d. R - (2/5)R = (3/5)R = 90,000 e. (1/5)R = 30,000 f. R = $150,000 (answer)

Car X and Y are traveling from A to B on the same route at constant speeds. Car X is initially behind Car Y, but Car X's speed is 1.25 times Car Y's speed. Car X passes Car Y at 1:30 pm. At 3:15 pm, Car X reaches B, and at that moment, Car Y is still 35 miles away from B. What is the speed of Car X

a. 35 = R(7/4) b. R = (4/7)35 = 20 c. X = 1.25y d. x-y= 20 e. 1.25y -y = 20 f. 0.25y = 20 g. y = 80 h 80 + 20 = 100 (X is going 100 mph-- answer)

Backsolving practice: A chemical supply company has 60 liters of a 40% H solution. How many liters of pure undiluted H must the chemists add so that the resultant solution is a 50% solution. A 12 B 15 C 20 D 24 E 30

a. 40% of 60 is 24, so the current ratio 24/30 b. Starting with answer choice C we get a 44/80. This is more than half, so we have added too much solution (C, D, and E will not work). c. Jumping to answer choice A, we get a 36/72 ratio. This is 50%, so A is the correct answer.

How many multiples of 8 are there from 200 to 640 inclusive.

a. 8x25 = 200 and 8x80 is 640 b. 200 is the 25th multiple of 8, 640 is the 80th multiple of 8, and both are included. c. number = 80 - 25 + 1 = 56 (answer)

Under optimal conditions, the V. Bacteria multiplies the size of its population by 5/2 every 4 hours. If there are 24 billion at 9:00 am., and optimal conditions are maintained, how many are there are 5:00 pm of the same day?

a. 9:00 am: 24 b b. 1:00 pm: 24(5/2)= 12(5)= 60 c. 5:00 pm: 60(5/2)= 30(5)= 150 b (answer)

Practice with D = RT a. What is the speed of someone who covers 240 miles in 6 hrs at a constant speed? b. How far does someone moving at 8 m/s move in 40 seconds? c. How much time does it take to move 300 feet at a speed of 20 ft/s?

a. = 40 mph b. = 320 meters c. = 15 seconds

Alice and Bruce each bought a refrigerator, and the sum of their purchases was 900. If twice of what Alice paid was 75 more than what Bruce paid what did Alice pay for her refrigerator?

a. A + B = 900 b. 2A = B+75 c. B= 2A-75 d. A + 2A -75= 900 e. 3A = 900 + 75 f. A = 300 + 25 g. A = $325 (answer)

Translation practice: a. A is 50% of B b. A is 50% greater than B

a. A = 0.5B b. A = 1.5B

At a certain school of 200 students, the students can study French, Spanish, both, or neither. Just as many study neither as study both. One quarter of those who study Spanish also study French. The total number who study French is 10 fewer than those who study Spanish only. How many students study French?

a. A = French; B= Both; C= Spanish; D= neither b. B=D c. 1/4(B+C)= B... B+C= 4B... d. C=3B e. A+B = C-10 f. A+B = 3B-10 g. A= 2B-10 h. A+B+C+D= 200 I. (2B-10) + B + 3B + B= 200 j. 7B = 210 k. B = 30 l. A = 2(30)-10 m. A = 50 students study French only (answer).

Wendy left the house with D dollars in cash. She made the following cash purchases: gas, for 3 less than 1/3 of D; a book, for 1/6 of D; stationary for six dollars more than 1/6 of D; and groceries, for 1/4 of D. After these four purchases, she had 4 left. What is the value of D?

a. D = (1/3)D - 3 + (1/6)D + (1/6)D + 6 + (1/4)D + 4 b. D = (1/3)D + (1/3)D + (1/4)D + 7 e. D= (4/12)D +(4/12)D + (3/12)D +7 f. D = 11/12D + 7 g. 7 = D - 11/12D h. 7 = 1/12D i. D = 84 (answer)

**Cassandra drove from A to B at a constant 60 mph speed. Shen then returned, on the same route, from B to A, at a constant speed of 20 mph. What was her average speed?

a. D = distance from a to b b. First leg: T = D/60 c. Second leg: T = D/20 d. Total time: T = D/20 + D/60 = 3D/20+D/60= 4D/60 = D/15 e. Total distance: 2D f. average speed= (2D)/(d/15) = (15/1D)(2D/1) = 30 mph (final answer)

Frank has 13 more dollars than Glenda does, and together they have 81. How much does Frank have.

a. F = 13 + G b. G + F = 81 c. G = F-13 d. F + F - 13 = 81 f. 2F = 94 g. F = 47 (answer)

***An airplane has a 3600 mile trip. It covers the first 1800 miles of a trip at 400 mph. Which of the following is the closest to the constant speed the plane would have to follow in the last 1800 miles so that the average speed of the whole trip is 450 mph. A. 450 B. 455 C. 500 D. 514 E. 600

a. First leg: 1800 = 400(t) b. First leg T = 4.5 c. Whole trip: 3600 = 450(t) d. whole trip: T = 8 e. Time for second leg: 8 - 4.5 = 3.5 hrs. f. Second leg 1800 = 3.5R g. Second leg: R = 1800/3.5 = 3600/7 = (3500/7) + (100/7) = 500 + approx 14 f. 514 mph = answer.

Bob drove 120 miles at 60 mph, then another 120 at 40 mph. What was he average speed for the total trip.

a. First leg: T = 120/60 = 2hrs. b. Second leg: T = 120/40 = 3hrs. c. Vavg = 240miles/5hrs = 48 mph (answer)

Which of the following could be true of at least some of the terms of the sequence defined by b sub n = (2n-1)(2n+3) I. Divisible by 2 II. Divisible by 3 III. Divisible by 5

a. II and III only b. Never I because it will always work out to be odd time odd, which always has an odd product. Odd numbers are never divisible by 2.

**Let T be a sequence of the form a sub n = a sub 1 + d(n-1). If a sub 3 = 17 and a sub 19 = 65, find a sub 10.

a. Let the initial term, a sub 1 = b, and let the common difference equal d. b. a sub 3 = b + 2d = 17 c. a sub 19 = b + 18d = 65 d. subtracting b from c we get 16d = 48; d = 3 e. plug 3 back into b and we get b = 11 f. Thus, a sub 10= 11 + 3(9) = 11 + 27 = 38 (answer)

Mark is 8 years older than Lisa, and Peter's age is 3 years less than three times Lisa's age. If the sum of their ages is 80, what is Peter's age?

a. M = L + 8 b. P = 3L - 3 c. L + M + P = 80 d. L + (L + 8) + (3L - 3) = 80 e. 5L + 5 = 80 f. 5L = 75 g. L = 15 (plug back into B for peter's age) h. P = 3(15) - 3 i. P= 42

Contract negotiations opened on the morning of March 20th, continued every day without a break, and ended late in the eventing of May 10th. For how many calendar days were contract negotiations in session.

a. March 20th to the 31s: 31 -20 + 1 = 12 days b. April = 30 c. In may, 10 days. e. 12 + 30 + 10 = 52 days.

***Martha and Paul started traveling from A to B at the same time. Martha traveled at a constant speed of 60 mph, and Paul at a constant speed of 40 mph. When Martha arrived at B, Paul was still 50 miles away. What is the distance between A and B?

a. Martha: D = 60(T) b. Paul: D-50 = 40(T) c. combined: 60T-50= 40T d. 60T= 40T+50 e. 20T= 50 f. T = 50/20= 5/2 (plug back into a for distance) g. D = 60(5/2) h. D= 30(5)= 150

Three criteria venn diagrams

a. Three groups with binary options b. there are 8 variables. c. start from the central region and work outward.

There are a total of 400 students at a school, which offers a chorus, baseball, and Italian. This year, 120 students are in the chorus, 40 students in both chorus and Italian, 45 students in both chorus & baseball, and 15 students do all three activities. If 220 students are in either Italian or baseball, then how many students are in none of the three activities.

a. Use a 3 circle venn diagram to solve. Depending on letter assignments, the answer will look something like the following b. T=400 c. B+C+E+F= 120 d. B+C= 40 e. C+F = 45 f. C=15 g. A+ B+ C+ D+ F+ G =220 h. B = 25; F= 30; E= 50 I. (A+ B+ C+ D+ F+ G)+ E= 270 j. H = 400 -270= 130 students who do none (answer)

Double Matrix problem In a company of 300 employees, 120 are females. a total of 200 employees have advanced degrees and the rest have a college degree only. If 80 employees are males with college degree only, how many are females with advanced degrees?

a. Use a box matrix with one more column and and one more row then there are categories to solve. b. 100 is the answer.

Double Matrix Problem: In a certain school, there are 80 freshmen, 100 sophomores, and 220 upperclassmen, drawn from three cities: A, B, and C. Sixty percent of students are from A, 30% from B, and the rest from C. All the students from C are freshmen. Half the students from B are upperclassman and the rest are split evenly between the other two grades. How many sophomores are from A?

a. Use a box matrix with one more column and and one more row then there are categories to solve. b. 70 students are sophomores from A (answer)

Pump X takes 28 hours to fill a pool. Pump Y takes 21 hours to fill the same pool. How long does it take them to fill the same pool if they are working simultaneously?

a. X = 1/28 b. Y = 1/21 c. 1/21 + 1/28 = 4/84 + 3/84 = 7/84 = 1/12 (12 hours to fill the pool).

For a sequence that has the same rule as the Fibonacci sequence, a sub n = a (sub n -1) + a (sub n -2) for n > 2, but the starting values of a sub 1 = 1 and a sub 2 = 3, find the value of a sub 6

a. a sub 1 = 1 and a sub 2 = 3 b. a sub 3 = 3 + 1 = 4 c. a sub 4= 4 + 3= 7 d. a sub 5 = 7 + 4 = 11 e. a sub 6 = 11 + 7 = 18

**14, 23, 32, 41, 50, 59 Find the 41st term of this sequence.

a. a sub 1 = 14; d = 9 b. a sub n = 14 + 9(n-1) c. a sub 41 = 14 + 9(41-1) d. a sub 41 = 14 + 9(40) = 14 + 360 = 374 (answer)

***Let S be the set of all positive integers that, when divided by 8, have a remainder of 5. What is the 76th number in this set?

a. a sub 1 = 5; d = 8 b. a sub n = 5 + 8(n-1) c. a sub 76 = 5 + 8(76-1) d. a sub 76 = 5 + 8(75) e. a sub 76 = 5 + 600 = 605 (answer)

For the sequence a sub n = 1/(n+2), find a sub 10 - a sub 6

a. a sub 10 = 1/(10+2) = 1/12 b. a sub 6 = 1/(6+2) = 1/8 c. a sub 10 - a sub 6 = 1/12 - 1/8 = 2/24 - 3/24 = -1/24 (answer)

Recursive sequences

a. a sub n - 1 is the term immediately before a sub n. b. If a sub n is defined in terms of a sub n - 1, this is a recursive definition, and the sequence is recursive. c. You can't figure out random terms in the sequence. You have to work from the first seed number all the way up to the value we want. d. on the test we will get either a sub n -1 or a sub n -2. If a sub n -2, we want to use the term before the term sitting immediately prior to n (two terms back).

Cars P & Q are approaching each other on the same highway. Car P is moving at 49 mph and Car Q is moving at 61 mph. At 2:00 pm, the are approaching each other and 121 mi apart. Eventually they pass each other. At what clock time are they moving away from each other and 44 miles apart.

a. add the gaps 121 + 44 = 165 b. add the speeds 61+49 = 110 c. formula: 165 = 110(T) d. T = 165/110 = 15/10 = 3/2 = 1.5 e. 2:00 + 1hr 30 min = 3:30 pm (answer)

How much HCl and how much water must we use to create 5 liters of 30% HCL solutions

a. amount of HCL= 0.3x5= 1.5 liters (answer 1) b. amount of water = 5 -1.5= 3.5 liters (answer 2)

***Suppose we start with 5 liters of a 30% HCl solution. How much water must we add to create a 20% solution.

a. amount of HCl: 0.3x5= 1.5 liters b. New solutions: 0.2(x)= 1.5 liters c. x = 1.5/.2 = 15/2 = 7.5 liters d. 7.5 - 5 = 2.5 liters of added water (answer)

**If b sub n = (b (sub n-1) -1)^2 +3, and b sub 1 = 1, find the value of b sub 4.

a. b sub 1 = 1 b. b sub 2 = (1-1) ^2 + 3= 3 c. b sub 3 = (3-1)^2 + 3 = 7 d. b sub 4 = (7-1)^2 + 3 = 39 (answer).

Of the 100 students in a school, 60 are in the band , and 35 are on the baseball team. If 25 students are neither in the band or on the baseball team, how many are in both?

a. create a venn diagram (A, B, C, D areas) b. A + B = 60 c. B+C= 35 d. D =25 e. A+B+C= 75 f. A + 35 = 75 (replacing B+C) g. A = 40 h. 40 + B = 60 i B = 20 students are in both (answer)