GMAT - Q - Equation Conversion

Working together, printer A and printer B would finish a task in 24 minutes. Printer A alone would finish the task in 60 minutes. How many pages does the task contain if printer B prints 5 pages a minute more than printer A?

*Combined rate:* 1/24 wpm *A's rate:* 1/60 wpm Solve for B's rate: >> 1/24 - 1/60 >> 5/120 - 2/120 >> 3/120 = 1.5/60 wpm Since the difference between B and A's output is 5 pages per minute... >> ...and: since B is 1.5 times faster than A >> ...thus: A's output should be 10 pages per minute and B's output should be 15 pages per minute >>....since: 15 - 10 = 5; and 10(1.5) = 15 Solve for number of pages in a task: >> ...since: combined time to complete is 24 minutes >> ...and: combined pages per minute is 25 pages >> ...thus: (25)(24) = 600 pages

Jack can paint a wall in 3 hours. John can do the same job in 5 hours. How long will it take if they work together?

*Jack's rate:* 1/3 work per hour *John's rate:* 1/5 work per hour Solve combined rate: >> 1/3 + 1/5 >> 5/15 + 3/15 >> 8/15 >> Thus, the combined time to complete 1 job is 15/8 hours

Working, independently X takes 12 hours to finish a certain work. He finishes 2/3 of the work. The rest of the work is finished by Y whose rate is 1/10 of X. In how much time does Y finish his work?

*X's rate:* 1/12 *Y's rate:* (1/12)(1/10) = 1/120 Since there is 1/3 of the work left, and Y's rate is 1/120 wph, thus... 1/120 * x = 1/3 >> (1/3) / (1/120) >> 1/3 * 120/1 >> 120/3 = 40 hours

Machine A and Machine B are used to manufacture 660 sprockets. It takes machine A ten hours longer to produce 660 sprockets than machine B. Machine B produces 10% more sprockets per hour than machine A. How many sprockets per hour does machine A produce?

>> Given that: *Machine A:* 660 = a * (t + 10) *Machine B:* 660 = b * t >> ...then: a = 660 / (t + 10) b = 660 / t >> ...also, since: (b) is 10% faster than (a) >> ...then: b = 660 / (t + 10) * 1.1 >> ...thus: b = 660 / (t + 10) * 1.1 = 660 / t Solve for (b) >> 660 / (t + 10) * 1.1 = 660 / t >> 660 / (t + 10) = (660 / t) * (1 / 1.1) >> 660 / (t + 10) = 660 / 1.1t >> t + 10 = 1.1t >> 10 = 1.1t - t >> 10 = 0.1t >> 10/0.1 = 100 = t Substitute (t = 100) into Machine A's rate: >> a = 660 / (100 + 10) >> a = 660 / 110 >> a = 6 sprockets per minute

The length of a rectangular garden surrounded by a walkway is twice its width. If the difference between the length and width of just the rectangular garden is 10 meters, what will be the width of the walkway if just the garden has a width of 6 meters?

>> Width of walkway: (x) >> ength of garden + walkway: (l) >> Width of garden + walkway: (w) >> Length of just garden: l - 2x >> Width of just garden: w - 2x = 6 >> Difference between length and width of just the garden: (l - 2x) - (w - 2x) = 10 >> ...thus: l - w = 10 >> Length of a garden surrounded by a walkway is twice its width: l = 2w >> Solve for (w): >> ...since: l = 2w >> ...and: l - w = 10 >> ...then: 2w - w = 10 >> ...thus: w = 10 >> Solve for (x): >> ...since: w = 10 >> ...and width of garden: w - 2x = 6 >> ...thus: 10 - 2x = 6 >> -2x = -4 >> x = 2

To qualify for a race, you need to average 60 mph driving two laps around a 1 mile long track. You have some sort of engine difficulty the first lap so that you only average 30 mph during that lap; how fast do you have to drive the second lap to average 60 for both of them?

Reconcile 2 DST (mile, mph, hr) equations; one for each lap *Lap 1:* 1 = 30 * (1/30) *Lap 2:* 1 = x * (1/x) *Total:* 2 = 60 * (2/60) Since we need to find the required speed for the second lap: (1/30) = (1/30) + (1/x); x = 0; thus, impossible

Two cyclists start at the same time from opposite ends of a course that is 45 miles long. One cyclist is riding at 14 mph and the second cyclist is riding at 16 mph. How long after they begin will they meet?

Reconcile 2 DST (mile, mph, hr) equations; one for each method of travel *Cyclist 1:* 14t = 14 * t *Cyclist 2:* 16t = 16 * t *Total:* 45 = n/a * n/a Solve for (x): >> 16t + 14t = 45 >> 30t = 45 >> t = 1.5 hours

An executive drove from home at an average speed of 30 mph to an airport where a helicopter was waiting. The executive boarded the helicopter and flew to the corporate offices at an average speed of 60 mph. The entire distance was 150 miles; the entire trip took three hours. Find the distance from the airport to the corporate offices.

Reconcile 2 DST (mile, mph, hr) equations; one for each method of travel *Driving:* (150 - x) = 30 * ((150 - x)/30) *Flying:* x = 60 * (x/60) *Total:* 150 = n/a * 3 Solve for (x): >> (150 - x)/30 + x/60 = 3 >> 150/30 - x/30 + x/60 = 3 >> 5 - x/30 + x/60 = 3 >> 5 - 2x/60 + x/60 = 3 >> 5 - x/60 = 3 >> 2 = x/60 >> x = 120 miles

A passenger train leaves the train depot 2 hours after a freight train left the same depot. The freight train is traveling 20 mph slower than the passenger train. Find the speed of the passenger train, if it overtakes the freight train in three hours.

Reconcile 2 DST (mile, mph, hr) equations; one for each method of travel *Freight:* d = (x - 20) * 5 *Passenger:* d = x * 3 *Total:* n/a = n/a * n/a NOTE 1: The freight train's time is 5 hours since this is the time it left the station until it was overtaken NOTE 2: (d) is the same for both, since they need to have traveled the same distance at the point that the passenger train overtakes Solve for (x): >> (x - 20)(5) = x(3) >> 5x - 100 = 3x >> 2x = 100 >> x = 50 mph

A boat travels for three hours with a current of 3 mph and then returns the same distance against the current in four hours. What is the boat's speed in calm water?

Reconcile 2 DST (mile, mph, hr) equations; one for each method of travel *Going:* (x + 3)3 = (x + 3) * 3 *Returning:* (x - 3)4 = (x - 3) * 4 *Total:* n/a = n/a * n/a NOTE: Going with a 3mph current means the current is adding 3mph to the boat's speed, while going against means the current is subtracting 3mph from the boat's speed Solve for (x): >> (x + 3)3 = (x - 3)4 >> 3x + 9 = 4x - 12 >> 12 + 9 = 4x - 3x >> x = 21 mph

In a consumer survey, 85% of those surveyed liked at least one of three products: 1, 2, and 3. 50% of those asked liked product 1, 30% liked product 2, and 20% liked product 3. If 5% of the people in the survey liked all three of the products, what percentage of the survey participants liked more than one of the three products?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 100 = (50 + 30 + 20) - (x + 5 + 5 + 5) + 5 >> 100 = 100 - x - 5 - 5 - 5 + 5 >> 100 = 100 - x - 10 >> 100 = 90 - x >> x = 10%

Of 20 Adults, 5 belong to A, 7 belong to B, and 9 belong to C. If 2 belong to all three organizations and 3 belong to exactly 2 organizations, how many belong to none of these organizations?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 20 = (5 + 7 + 9) - (3 + 2 + 2 + 2) + 2 + x >> 20 = 21 - 9 + 2 + x >> 20 = 14 + x >> x = 6 adults

In a class of 50 students, 20 play Hockey, 15 play Cricket and 11 play Football. 7 play both Hockey and Cricket, 4 play Cricket and Football and 5 play Hockey and football. If 18 students do not play any of these given sports, how many students play exactly two of these sports?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 50 = (20 + 15 + 11) - (7 + 4 + 5) + y + 18 >> 50 = 46 - 16 + y + 18 >> 50 = 48 + y >> y = 2 Since we now know that 2 students play all three sports, we need to subtract this 2 from (7 + 4 + 5) three times: >> x = 7 + 4 + 5 - 2 - 2 - 2 >> x = 10 students

There are 50 employees in the office of ABC Company. Of these, 22 have taken an accounting course, 15 have taken a course in finance and 14 have taken a marketing course. Nine of the employees have taken exactly two of the courses and 1 employee has taken all three of the courses. How many of the 50 employees have taken none of the courses?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 50 = (22 + 15 + 14) - (9 + 1 + 1 + 1) + 1 + x >> 50 = 51 - 12 + 1 + x >> 50 = 40 + x >> x = 10 employees

Each of the 59 members in a high school class is required to sign up for a minimum of one and a maximum of three academic clubs. The three clubs to choose from are the poetry club, the history club, and the writing club. A total of 22 students sign up for the poetry club, 27 students for the history club, and 28 students for the writing club. If 6 students sign up for exactly two clubs, how many students sign up for all three clubs?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 59 = (22 + 27 + 28) - (6 + x + x + x) + x + 0 >> 59 = 77 - 6 - x - x - x + x + 0 >> 59 = 71 - 2x + 0 >> 2x = 71 - 59 = 12 >> x = 6 students The reason for (6 + AnBnC + AnBnC + AnBnC) is because the (AnB + AnC + BnC) portion of the equation actually contains the 3 overlaps three times in addition to each of the three 2 overlaps; however, the question only gives us the total of the 2 overlaps (6) and no information regarding the three overlaps, thus we must add the three AnBnC back into that portion of the equation

In the city of San Durango, 60 people own cats, dogs, or rabbits. If 30 people owned cats, 40 owned dogs, 10 owned rabbits, and 12 owned exactly two of the three types of pet, how many people owned all three?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 60 = (30 + 40 + 10) - (12 + x + x + x) + x >> 60 = 80 - 12 - x - x - x + x >> 60 = 68 - 2x >> 2x = 8 >> x = 4 people

This semester, each of the 90 students in a certain class took at least one course from A, B, and C. If 60 students took A, 40 students took B, 20 students took C, and 5 students took all the three, how many students took exactly two courses?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> 90 = (60 + 40 + 20) - (x + 5 + 5 + 5) + 5 + 0 >> 90 = 120 - x - 5 - 5 - 5 + 5 >> 90 = 120 - x - 10 >> 90 = 110 - x >> x = 20 students

When Professor Wang looked at the rosters for this term's classes, she saw that the roster for her economics class (E) had 26 names, the roster for her marketing class (M) had 28, and the roster for her statistics class (S) had 18. When she compared the rosters, she saw that E and M had 9 names in common, E and S had 7, and M and S had 10. She also saw that 4 names were on all 3 rosters. If the rosters for Professor Wang's 3 classes are combined with no student's name listed more than once, how many names will be on the combined roster?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> x = (26 + 28 + 18) - (9 + 7 + 10) + 4 >> x = 72 - 26 + 4 >> x = 50 students

Workers are grouped by their areas of expertise, and are placed on at least one team. 20 are on the marketing team, 30 are on the Sales team, and 40 are on the Vision team. 5 workers are on both the Marketing and Sales teams, 6 workers are on both the Sales and Vision teams, 9 workers are on both the Marketing and Vision teams, and 4 workers are on all three teams. How many workers are there in total?

Remember: Total = (A + B + C) - (AnB + AnC + BnC) + (AnBnC) + Neither So... >> (20 + 30 + 40) - (5 + 6 + 9) + 4 + 0 >> 90 - 20 + 4 >> 74 total workers

DS: A student has decided to take GMAT and TOEFL examinations, for which he has allocated a certain number of days for preparation. On any given day, he does not prepare for both GMAT and TOEFL. How many days did he allocate for the preparation? (1) He did not prepare for GMAT on 10 days and for TOEFL on 12 days. (2) He prepared for either GMAT or TOEFL on 14 days

Remember: Total = (A + B) - (AnB) + Neither[if applicable] Base equation: >> Total = GMAT + TOEFL + Neither >> x = (g + t) + y With Statement 1: >> x - g = 10 >> x - t = 12 ---> NOT SUFFICIENT With Statement 2: >> x = 14 + y ---> NOT SUFFICIENT With Both Statements: >> x - g = 10... thus: x - 10 = g >> x - t = 12... thus: x - 12 = t >> x - y = 14... thus: x - 14 = y >> Substitute: >> x = (x - 10 + x - 12) + x - 14 >> x = 2x - 22 + x - 14 >> x = 3x - 36 >> 36 = 2x >> x = 18

A friend visits you from 100 miles away and tells you that it took him 4 hours to reach. However, he was stuck in traffic along the way and could only travel 50 mph whenever he could move. How many hours was your friend stuck in traffic?

Time spent driving 50 mph: (y) Speed when stuck in traffic: 0 mph >> 100 = 50(y) + 0(4 - y) >> 100 = 50(y) >> 2 = (y) >> 4 - 2 = 2

The length of a rectangular garden is 2 meters less than its width. Express its length in terms of its width

l = w - 2

The ratio of the length of a rectangular garden to its width its 2. Express its length in terms of its width

l = w(2)

Mary's age is 10 more than twice that of Jim's

m = 2j + 10