Magoosh - Math - Word problems

Cassandra drove from A to B at a constant 60 miles an hour speed, then she returned on the same route, from B to A at a constant speed of 20 miles an hour, what was their average speed for the whole trip?

Average speed D = RT Example Cassandra drove from A to B at a constant 60 miles an hour speed, then she returned on the same route, from B to A at a constant speed of 20 miles an hour, what was their average speed for the whole trip? First half + second half = total D/60 + D/20 = 2D/x D/60 + 3D/60 = 2D/x 4D/60 = 2D/x, x = 30 Alternate approach Pick a random distance with easy math D = 60 First half + second half = total 60/60 + 60/20 = 120/x 1 + 3 = 120/x, x = 30

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

Growth and decay 12 days are 4 intervals of 3 96 —> 48 —> 24 —> 12

Bacteria X multiplies the size of its population by 5/2 every 4 hours. If there are 24 billion at 9 a.m., and optimal conditions are maintained, how many are there at 5 p.m. of the same day?

Growth and decay 9 am to 5 pm = 2 intervals of 4 24 x 5/2 x 5/2 = 150 billion

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

Inclusive counting We use inclusive counting when both endpoints (the starting value and ending value) are included in the counting. Essentially must add 1 to the subtracted difference Example Contract negotiations opened on the morning of March 20th, continued every day without a break and ended in the evening of May 10th. For how many calendar days were contract negotiations in session? Part of March + April + part of May = total (31 - 20 + 1) + 30 + (10 - 1 + 1) = total 12 + 30 + 10 = 52 How many multiples of 8 are there from 200 to 640 inclusive? 8*25 = 200 and 8*80 = 640 80 - 25 + 1 = 56

Let T be a sequence of the form, a3 = 17, a19 = 65, find a10.

Arithmetic sequence Finding a formula of a sequence of numbers an = a1 + d(n-1) Example: 5, 12, 19, 26, etc. an = 5 + 7(n-1) D = common difference → an = 5 + d(n-1) d R = fixed remainder → an = a1 + d(n-1) Example Let T be a sequence of the form, a3 = 17, a19 = 65, find a10. 17 = a1 + x(2) → a1 = 2x -17 65 = a1 +x(18) → a1 = 18x - 65 2x - 17 = 18x - 65 65 - 17 = 18x - 2x 48 = 16x x = 3 a1 = 17 - 3(2) = 11 a10 = 11 + 3(9) = 38

14, 23, 32, 41, 50, 59. Find the 41st term

Arithmetic sequence Finding a formula of a sequence of numbers an = a1 + d(n-1) Example: 5, 12, 19, 26, etc. an = 5 + 7(n-1) D = common difference → an = 5 + d(n-1) d R = fixed remainder → an = a1 + d(n-1) Example 14, 23, 32, 41, 50, 59. Find the 41st term an = a1 + d(n-1) an = 14 + 9(n-1) A41 = 14 + 9(40) = 14 + 360 = 374

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

Arithmetic sequence Finding a formula of a sequence of numbers an = a1 + d(n-1) Example: 5, 12, 19, 26, etc. an = 5 + 7(n-1) D = common difference → an = 5 + d(n-1) d R = fixed remainder → an = a1 + d(n-1) Example Let S be the set of all positive integer that, when divided by 8, have a remainder of 5. What is the 76th number in this set? an = a1+ d(n-1) an = 5 + 8(n-1) = 5 + 8(75) = 5 + 600 = 605

An airplane has a 3600 mile trip. It covers the first 1800 miles of the trip at 400 miles an hour. Which of the following is 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 miles per hour?

Average speed Example An airplane has a 3600 mile trip. It covers the first 1800 miles of the trip at 400 miles an hour. Which of the following is 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 miles per hour? First half + second half = total 1800/400 + 1800/x = 3600/450 4.5 + 1800/x = 8 1800/x = 3.5 x = 1800 x 2/7 = 3600/7 = 514.2

What are the two techniques of backsolving?

Backsolving If all answer choices are numbers, can you backsolving technique If you trial middle choice (i.e. usually choice C), can usually element half to 3 answers at once Alternate strategy Trial B = rule out A and B = 40% of choices Then trial D

Using backsolving A chemical supply company has 60L of a 40% solution of HNO3 solution. How many liters of pure undiluted acid must the chemists add so that the resultant solution is a 50% solution? A = 12 B = 15 C = 20 D = 24 E = 30

Backsolving If all answer choices are numbers, can you backsolving technique If you trial middle choice (i.e. usually choice C), can usually element half to 3 answers at once Alternate strategy Trial B = rule out A and B = 40% of choices Then trial D A chemical supply company has 60L of a 40% solution of HNO3 solution. How many liters of pure undiluted acid must the chemists add so that the resultant solution is a 50% solution? A = 12 B = 15 C = 20 D = 24 E = 30 Choose c = 60L x 0.4 + 20L x 1.0? = 0.5 x 80 → 24 + 20 = 40, incorrect, smaller Choose a = 60L x 0.4 + 12L x 1.0? = 0.5 x 72 → 24 + 12 = 36, correct

Using backsolving In a certain state, schools pay 2% tax on food and 8% tax on stationery. The 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 spent on food? A = 200 B = 250 C = 300 D = 350 E = 400

Backsolving If all answer choices are numbers, can you backsolving technique If you trial middle choice (i.e. usually choice C), can usually element half to 3 answers at once Alternate strategy Trial B = rule out A and B = 40% of choices Then trial D Example In a certain state, schools pay 2% tax on food and 8% tax on stationery. The 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 spent on food? A = 200 B = 250 C = 300 D = 350 E = 400 Choose c = 300 (1.02) + 200 (1.08) = 522 → 22 tax too much Chose e = 400 (1.02) + 100 (1.08) = 516 → 16 tax too little Answer must be D = 350 Conventional way: x(1.02) + (500-x)(1.08) = 500 + 19 1.02x + 540 - 1.08x = 519 540 - 519 = 0.06x 21 = 0.06x X = 350

If n is an integer greater than 50, then the expression (n*2-2n)(n+1)(n-1) MUST be divisible by which of the following? I. 8, II. 12, III. 18. Answers (A) = I only, (B) = II only, (C) = I and II only, (D) = II and III only, (E) = I, II, III

Consecutive integers {1,2,3} or {-4,-5,-6} Example If n is an integer greater than 50, then the expression (n*2-2n)(n+1)(n-1) MUST be divisible by which of the following? I. 8, II. 12, III. 18. Answers (A) = I only, (B) = II only, (C) = I and II only, (D) = II and III only, (E) = I, II, III = n(n-2)(n-1)(n+1) = (n-2)(n-1)(n)(n+1) 51 x 52 x 53 x 54 Has at least one multiple of 4 and one multiple of 3 Answer is C

Fact of consecutive integers A set of n consecutive integers will always contain one number __________ by n

Consecutive integers {1,2,3} or {-4,-5,-6} Fact 1: A set of n consecutive integers will always contain one number divisible by n 20,21,22,23,24,25,27; 7 integers, will always have one number divisible by 7 Fact 2: If n is odd, the sum of a set of n consecutive integers will always be divisible by n. 20,21,22,23,24,25,27; sum = 161 is divisible by 7 Fact 3: In a set of consecutive integers: 2 odd, 1 even or 2 evens, and 1 odd. In a set of 4 integers: 2 odd, 2 evens Fact 4: Test will usually give consecutive integers as an algebraic problem {n, n+1, n+2, n+3}

Fact of consecutive integers If n is odd, the __________ of a set of n consecutive integers will always be _________ by n.

Consecutive integers {1,2,3} or {-4,-5,-6} Fact 1: A set of n consecutive integers will always contain one number divisible by n 20,21,22,23,24,25,27; 7 integers, will always have one number divisible by 7 Fact 2: If n is odd, the sum of a set of n consecutive integers will always be divisible by n. 20,21,22,23,24,25,27; sum = 161 is divisible by 7 Fact 3: In a set of consecutive integers: 2 odd, 1 even or 2 evens, and 1 odd. In a set of 4 integers: 2 odd, 2 evens Fact 4: Test will usually give consecutive integers as an algebraic problem {n, n+1, n+2, n+3}

Fact of consecutive integers In a set of consecutive integers: ___ odd, ___ even or ___ evens, and ___ odd. In a set of 4 integers: ___ odd, ___ evens

Consecutive integers {1,2,3} or {-4,-5,-6} Fact 1: A set of n consecutive integers will always contain one number divisible by n 20,21,22,23,24,25,27; 7 integers, will always have one number divisible by 7 Fact 2: If n is odd, the sum of a set of n consecutive integers will always be divisible by n. 20,21,22,23,24,25,27; sum = 161 is divisible by 7 Fact 3: In a set of consecutive integers: 2 odd, 1 even or 2 evens, and 1 odd. In a set of 4 integers: 2 odd, 2 evens Fact 4: Test will usually give consecutive integers as an algebraic problem {n, n+1, n+2, n+3}

N = 135 is the lowest number in a set of 41 consecutive multiples of 5. What is the difference between the lowest and highest numbers in the set?

Consecutive integers {1,2,3} or {-4,-5,-6} N = 135 is the lowest number in a set of 41 consecutive multiples of 5. What is the difference between the lowest and highest numbers in the set? 40 x 5 = 200 + 135 = 335 335 - 135 = 200

Which of the following could be true of at least some of the terms of the sequence defined by bn=(2n-1)(2n+3). I - divisible by 2, II - divisible by 3, III - divisible by 5. Choices (A) I only, (B) II only, (C) I and II only, (D) II and III only, (E) I, II, III

Introduction of sequences an = sequence, a5 = fifth term = 28 Rn =n(n+2) N = 1, r1 = 1(3) = 3 N = 2, r2 = 2(4) = 8 N = 3, r3 = 3(5) = 15 N = 4, r4 = 46) = 24 Example an = 1/n+2, find a10 - a6 A10 = 1/10+2 - 1/6+2 = 1/12 - 1/8 = 2/24 - 3/24 == -1/24 Which of the following could be true of at least some of the terms of the sequence defined by bn=(2n-1)(2n+3). I - divisible by 2, II - divisible by 3, III - divisible by 5. Choices (A) I only, (B) II only, (C) I and II only, (D) II and III only, (E) I, II, III N = 1, b = 1x5 = 5 N = 2, b = 3x7 = 21 Nothing is even, thus divisible by 2 Answer is D

an = 1/n+2, find a10 - a6

Introduction of sequences an = sequence, a5 = fifth term = 28 Rn =n(n+2) N = 1, r1 = 1(3) = 3 N = 2, r2 = 2(4) = 8 N = 3, r3 = 3(5) = 15 N = 4, r4 = 46) = 24 Example an = 1/n+2, find a10 - a6 A10 = 1/10+2 - 1/6+2 = 1/12 - 1/8 = 2/24 - 3/24 == -1/24 Which of the following could be true of at least some of the terms of the sequence defined by bn=(2n-1)(2n+3). I - divisible by 2, II - divisible by 3, III - divisible by 5. Choices (A) I only, (B) II only, (C) I and II only, (D) II and III only, (E) I, II, III N = 1, b = 1x5 = 5 N = 2, b = 3x7 = 21 Nothing is even, thus divisible by 2 Answer is D

Suppose we start with unlimited supplies of a 20% HCl solution and a 50% HCl solution. We combine X liters of the first with Y liters of the second to produce seven liters of a 40% solution. What does X equal?

Mixture problems 0.2x + 0.5(7-x) = 0.4 x 7 0.2x + 3.5 - 0.5x = 2.8 2x + 35 - 5x = 28 -3x = -7 x = 7/3 = 3 2/3L

We have 8L of 60% HCl. We add 4 L of c% HCl and the result is 12L of 50% HCl. What is C?

Mixture problems 0.6 x 8 + c x 4 = 12 x 0.5 4c = 6 - 4.8 = 1.2 c = 0.3 or 30% Alternate Solute in 1st solution: 0.6 x 8 = 4.8L Solute in final solution: 0.5 x 12 = 6L Solute in 2nd solution: 6L - 4.8L = 1.2L Concentration of 2nd solution: 1.2L/4L = 3/10 = 30%

How much HCL and how much water must we use to create 5 L of a 30% HCl solution

Mixture problems Amount of 100% HCl = 0.3 x 5L = 1.5 L of 100% HCl Amount of water = 5L - 1.5L = 3.5L

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

Mixture problems Solute in 1st solution: 0.3 x 5 = 1.5L Volume of final solution: 0.2 x = 1.5L = x 1.5L/0.2 = 7.5L

Kevin drove from A to B at a constant speed of 60 miles an hour, turned around and returned at a constant speed of 80 miles an hour. Exactly 4 hours before the end of his trip, he was still approaching B, and only 15 miles away from it. What is the distance between A and B.

Multiple travelers D = RT A—----------------P>----15----> A<-----------------------------B Time to go 15 miles at 60 mph = ¼ hour Time to go B to A = 4 - ¼ = 3 ¾ D = 80(3 ¾ ) = 80 x 15/4 = 300

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 is A to B?

Multiple travelers D = RT D - 60 = RT (Frank) D + 60 = 1.5RT (Georga) RT + 60 = 1.5RT - 90 150 = 0.5RT 300 = RT

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

Multiple travelers D = RT, Martha distance, M = 60T, Paul distance, M-50 = 40T M/60 = M-50/40 40M = 60M - 300 20M = 300, M = 150 miles Alternate D = RT, Martha distance, D = 60T, Paul distance, D-50 = 40T D-50 = 40T 60T - 50 = 40T 20T = 50, T = 2.5 D = 60 x 2.5 = 150 miles

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

The double matrix method Example In a company of 300 employees, 120 are females. A total of 200 employees have advanced degrees, and the rest have college degree only. If 80 employees are males with college degrees only, how many are females with advanced degrees? Some pieces of information leads to others Female Male TOTALS College A C E Advanced B D F TOTALS G H T Female Male TOTALS College A 80 E Advanced B D 200 TOTALS 120 H 300 Female Male TOTALS College (20) 80 100 Advanced (100) (100) 200 TOTALS 120 (180) 300 Answer = 100

How to solve of recursive sequences

Recursive sequences With recursive sequence, no way to jump to calculate value. Instead, must find each term in sequence up to the desired term

A sequence is defined by sn = (sn-1- 1)(sn-2) for n > 2, s1 = 2 and s2 = 3. Find the value of s6

Recursive sequences With recursive sequence, no way to jump to calculate value. Instead, must find each term in sequence up to the desired term A sequence is defined by sn = (sn-1- 1)(sn-2) for n > 2, s1 = 2 and s2 = 3. Find the value of s6 s3 = (sn-1- 1)(sn-2) = 3-1 x 2 = 4 s4 = . . . . . . . . . . . . . = 4-1 x 3 = 9 s5 = . . . . . . . . . . . . . = 9-1 x 4 = 32 s6 = . . . . . . . . . . . . . = 32-1 x 9 = 279

Solve this sequence 1,1,2,3,5,8,13,21,24,34,55,89

Recursive sequences With recursive sequence, no way to jump to calculate value. Instead, must find each term in sequence up to the desired term Basic: an = a1 + an-1 Fibonacci sequence 1,1,2,3,5,8,13,21,24,34,55,89 2 = 1 + 1; 3 = 1 + 2; 5 = 2 + 3; 8 = 3 + 5, etc. an = an-1 + an-2

If bn = (bn-1 - 1)*2 + 3, b1 = 1, find b4

Recursive sequences With recursive sequence, no way to jump to calculate value. Instead, must find each term in sequence up to the desired term Example If bn = (bn-1 - 1)*2 + 3, b1 = 1, find b4 b1 = (0-1)*2 + 3 = 1 b2 = (1-1)*2 + 3 = 3 b3 = (3-1)*2 + 3 = 7 b4 = ( 7-1)*2 + 3 = 39

100 students in a school, 60 are in the band (A), 35 are on the baseball team(C). If 25 students are in neither band or baseball team(D), how many in both(B)?

Sets and Venn diagrams A + B + C + D = Total Example 100 students in a school, 60 are in the band (A), 35 are on the baseball team(C). If 25 students are in neither band or baseball team(D), how many in both(B)? A + B = 60 or A = 60 - B B + C = 35 or C = 35 - B D = 25 A + B + C + D = 100 (60 - B) + B + (35 - B) + 25 = 100 120 - B = 100 B = 20

Shrinking or expanding gaps Case 1 ----------> ---> (add or subtract)

Shrinking or expanding gaps Add

Shrinking or expanding gaps Case 1 ------> <------ (add or subtract)

Shrinking or expanding gaps Add

Shrinking or expanding gaps Case 1 ---> ----------> (add or subtract)

Shrinking or expanding gaps Add

Shrinking or expanding gaps Case 1 <------ ------> (add or subtract)

Shrinking or expanding gaps Add

A car and truck are moving in the same direction on the highway. The truck is moving at 50 miles an hour and the car is travelling at a constant speed. At 3:00 PM, the car is 30 miles behind the truck and at 4:30, the car overtakes and passes the truck. What is the speed of the car?

Shrinking or expanding gaps D = RT 30 = (x-50)1.5 30 = 1.5x - 75 105 = 1.5x 70 = x Alternate Gap: R = D/T = 30/1.5 = 20 mph Car = Truck + 20 = 50 + 20 = 70 mph

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

Shrinking or expanding gaps D = RT Gap = 49 + 61 = 110 mph shrinking Distance = 121 + 44 = 162 miles Solve for time D = RT 165 = 110T 33 = 22T 3/2 = T = 1.5 hours 1.5 hours + 2:00 = 3:30

City J is 480 miles North of City K. At 10:00 AM, Car M starts driving South and Car N starts in K, driving North, and it's going twice as fast as M. Both cars maintain constant speeds and they pass each other at 2:00 PM going in opposite directions. What is the speed of car N?

Shrinking or expanding gaps D = RT, D = 480, R = x + 2x, T = 4 480 = (x + 2x)4 480 = 12x 40 = x Car N = 2x = 80 Alternate Gap: R = D/T = 480/4 = 120 mph M + N = 120 x + 2x = 120 x = 40, Car N = 2x = 80

When Amelia and Brad detail a car together, one car takes 3 hours to detail. When Amelia details the car alone, one car takes 4 hours. How long does it take Brad, working alone, to detail one car?

Work questions 1/A + 1/B = 1/A&B 1/4 + 1/B = 1/3 1/B = 1/3 - 1/4 → 1/12 → B = 12 hours

What is the formula for work

Work questions A = RT Combining two workers A/R + A/R = Total A/total R 1/r + 1/R = 1/r+R A is work done, R = rate of work done, T = time R = A/T or T = A/R

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

Sums of sequences Example What is the sum of all the multiples of 20 from 160 to 840 inclusive? 160/20 = 8, 840/20 = 42, 35 multiples, number pairs = 17.5 median 160 + 168 . . .(500) . . . 832 + 840 17 pairs of 1000 + 500 = 17500 or (160 + 840) 35/2 = 17500

Sum of even integers to 100

Sums of sequences Sum of even integers = (a1 + an) x n/2 2 + 4 + 6 . . . . 96 + 98 + 100 25 pairs of 102 → 2550 (2 + 100) x 50/2 = 2550

What is sum of 1 to 100

Sums of sequences What is sum of 1 to 100 1 + 2 + 3 + 4 + . . . . . 97 + 98 + 99 + 100 50 pairs of 101 = 101 x 50 = 5050 Sum of consecutive integers = (n+1) x n/2

What is the sum of all the multiples of 5 that are greater than 100 and less than 200

Sums of sequences What is the sum of all the multiples of 5 that are greater than 100 and less than 200 (105 + 195) 19/2 = 300 x 9.5 = 2850

In a certain school, there are 80 freshmen, 100 sophomores and 220 upperclassmen, drawn from three cities. 60% of students from A, 30% from B, rest from C. Half of the students from B are upperclassmen and the rest are evenly split between the other two grades. How many sophmores are from A?

The double matrix method In a certain school, there are 80 freshmen, 100 sophomores and 220 upperclassmen, drawn from three cities. 60% of students from A, 30% from B, rest from C. Half of the students from B are upperclassmen and the rest are evenly split between the other two grades. How many sophmores are from A? Fresh Soph Upper TOTALS City A 60% City B 25% 25% 50% 30% City C TOTALS 80 100 220 Fresh Soph Upper TOTALS City A (70) 60% (= 240) City B (30) (30) 50% (=60) 30% (= 120) City C (40) 0 0 (10% = 40) TOTALS 80 100 220 (400) Answer = 70

There are a total of 400 students at a school and this school offers a chorus, baseball, and Italian. This year 120 students are in the chorus, 40 students are in both chorus and Italian, 45 students are in both chorus and 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?

Three criteria venn diagrams Example There are a total of 400 students at a school and this school offers a chorus, baseball, and Italian. This year 120 students are in the chorus, 40 students are in both chorus and Italian, 45 students are in both chorus and 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? Draw 3 criteria Venn diagram T = 400 Italian = A + B + C + D Chrous = B + C + E + F Baseball = C + D + F + G Italian + Baseball = C + D Italian + Chorus = C + B Chorus + Baseball = C + F All 3 = C None = H Italian or baseball = A + B + C + D + F + G = 220 B = 25; F = 30; E = 50 → A + B + C + D + F + G + E = 270 H = 400 - 270 = 130

What are 4 guidelines for picking numbers for VICs?

VICs: Picking numbers Sometimes easier to pick numbers to solve problem Guideline for picking numbers Avoid 0 or 1 Pick numbers for different variables, and numbers different from those in the given problem. Don't pick multiples that are multiples of each other; pick prime numbers Keep numbers small and that are easy to calculate

Two ideas of VICs for algebraic approach

VICs: an algebraic approach First idea: because using algebra, doesn't mean you have to think through number Second idea: more than one way to write an expression and thus does not match up with answer choices

Two approaches to VICs?

VICs: an introduction Some word problems present all answer choices in terms of variables Variable In the answer Choices: VICs 2 general approaches Algebriac approach Simply picking numbers to fit terms Mainly good for eliminating numbers, not choosing the correct numbers

Jennifer can buy watches at a price of B dollars per watch, which she marks up buy a certain percentage before selling. If she makes a total profit of T by selling N watches, then in terms of B and T and N, what is the percent of the markup from her buy price to her sell price? A = 100T/(NB) B = TB/100N C = 100TN/B D = ((T/N) - B)/100B E = 100(T-NB)N

VICs: an introduction Some word problems present all answer choices in terms of variables Variable In the answer Choices: VICs 2 general approaches Algebriac approach Simply picking numbers to fit terms Mainly good for eliminating numbers, not choosing the correct numbers Example Jennifer can buy watches at a price of B dollars per watch, which she marks up buy a certain percentage before selling. If she makes a total profit of T by selling N watches, then in terms of B and T and N, what is the percent of the markup from her buy price to her sell price? A = 100T/(NB) B = TB/100N C = 100TN/B D = ((T/N) - B)/100B E = 100(T-NB)N Choose B = 10, N = 20, T = 60; percent = 30 A works, B = doesn't work, C = doesn't work, D = doesn't work, E = doesn't work. Got lucky with one choice.

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?

Work questions 1/A + 1/B = 1/A&B 1/28 + 1/21 = 1/x 3 + 4/84 = 1/x 7/84 = 1/x 12 = x