Average, Work/Rate

Question 1

Average Formula:

Answer 1

Sum of Terms/Number of Terms = Average

E.g. if sum of ages of 4 siblings is 84, then:

Average = 84/4 
Average = 21

Question 2

Distance/Work Formula:

Answer 2

Distance = Rate x Time

E.g. D = 40mph x 4 h
D = 160m

Work = Rate x Time

Question 3

Average Speed Formula:

Answer 3

Average Speed = Total Distance/Total time

E.g. if a car drives 100 miles in 2 hours:

Average speed = 100/2 = 50 miles per hour.

REMEMBER: Rate is always counted in distance per hour (d/t where t is time counted in hours). That means if you are given minutes you have to convert that number into a number per hour. E.g. 3 mins is 1/20 of an hour so use 1/20 for t.

Question 4

Average Formula With Odd and Even Sequences:

Answer 4

A fraction where the sum of a sequence of consecutive integers is divided by the number of integers is the average formula:

E.g. 1 + 2 + 3 + 4 + 5/5

Rule for Results:

If the number of integers in the Average formula is odd, the result of the division will be an integer.

If the number of integers in the Average formula is even, the result cannot be an integer but will be a fraction.

Question 5

Rate Formula:

Answer 5

Quantity A/Quantity B

Question 6

Balanced Average:

Answer 6

You know the Average Formula: Sum of Values/Number of values. You can save time on doing calculations by thinking of the average as the “balancing point” between the values, of which some are above and others below the average.

E.g. Average of 43, 44, 45 and x is 45. What is x?

You can see quickly that if 45 is the average then 43 is 2 below average and 44 is 1 below average. So in order to still get 45 as the average x has to be 3 above average, i.e. 48.

I.E. The amount above the average must equal the amount below the average.

Question 7

Solving Complicated Average Questions:

Answer 7

Instead of setting up complicated algebraic equations and expressions, pick very simple and permissible numbers and most of the questions can be solved with very simple arithmetic.

Question 8

Feet per Mile:

Answer 8

There are 5280 feet per mile.

Question 9

Inches per Feet:

Answer 9

There are 12 inches in one feet.

Question 10

Rate Formula:

Answer 10

Anything with the word PER is a rate. A rate is any quantity of A per quantity of B. Formula:

Rate A per B = Quantity A/Quantity B

E.g. $2 per gallon would be $2/gallon.

Question 11

Solving Rate Problems:

Answer 11

There are 2 ways to solve rate problems:

1. Algebraically
2. Through picking numbers (pick easy numbers).
3. Algebraically:
   Set up the equation so that units cancel out.

E.g. A and B are 3471 miles apart from each other. How much are A and B apart from each other in inches? 5280 feet are 1 mile. 12 inches are in 1 foot.

Equation:

3471 miles x 5280 feet/mile x 12 inches/foot

You can cancel the units “feet” and “miles” so that you are left with:

3471 x 5280 x 12 inches

Question 12

Average Rates and Speed in Multi-Part Journeys:

Answer 12

Remember that if you get the average rates or speed of a multi-part journey you can’t just split the difference of the average of the whole journey between the two parts. Just as you cannot use the averages of two different portions of a group to determine the group’s overall average (unless you know the weights of the portions), you also cannot find the average speed of a journey from the speeds of two parts unless you know the proportions of time spent or distance traveled in those parts.

You need to use the Average Rate Formula:

Average rate A per B: Total A/Total B

E.g. John travels 30 miles in 2 hours and then 60 miles in 3 hours. What is his average speed in mph?

Average Speed = Total Miles/Total Hours

30+60/2+3 = 90/5 = 18 miles per hour.

It makes sense that the rate falls between the first rate (15mph) and second rate (20mph). And because John spent more time traveling at 20mph the overall average rate will be closer to that rate than 15mph. Seeing this quickly also lets you eliminate incorrect answers quickly.

Question 13

Organizing Multi-Part Journey Problems:

Answer 13

There’s a lot of information in these questions. Organize your work in a chart like this:

                      Rate                  Time              Distance  Part 1 Part 2 Entire Trip 

Put a question mark in the box of the value you’re looking for. Then you can put in the info already given to you and calculate the rest by these rules:

Rate x Time = Distance (multiply rows across)
Time 1 + Time 2 = Time of entire trip (time boxes add down)
Distance 1 + Distance 2 = Distance of entire trip (add down)

BUT:
Speed rates DO NOT add down. They are calculated by dividing distance by time. Use the formula to calculate the average speed of the entire trip by adding the two distances and dividing them by the sum of the two times.

Average Speed: Distance1 + Distance 2/Time 1 + Time 2

Missing information in the chart:

You need information on at least two things in the three part equation, rate, distance, and time to solve for the missing one. You don’t need actual numbers, though. Values like fractions or percentages that show relationships can be enough.

Distance: You don’t always need the absolute distance. If you are not given values for the distance but given some information about the distance, like it’s the same in both parts, you can PICK NUMBERS. Pick easy numbers. You can also set up equations and solve that way.

Time: You don’t always need the absolute time. If you are not given the time but some information about the time, e.g. fractions, you can set up equations and solve.

Question 14

Combined Work Rates:

Answer 14

Combined work questions give you information about different people or machines that can perform the same job in different amounts of time.

Rate = Number of Tasks/Time to Complete Tasks

But on GMAT they won’t give you the rates at which people or machines work but instead the time it takes for each individual to complete the work.

To calculate the time needed for everyone working together to finish the job:

Combined Work Formula:

1/Total Time = 1/A + 1/B + 1/C….+1/N

Where A, B, C etc. are the time it takes the individual people or machines to do the jobs by themselves. The value “1” in the numerator represent the task. The tasks have to be the same for this formula to work of course. And the value “1” can stand for any task (e.g. making 1000 nails)

So essentially you are just using the rate formula here and adding the different rates:

Rate = Task/Time Total Rate = Rate X + Rate Y

REMEMBER: This formulas only work if you are given the times and they are all for the same-sized task (e.g. all making 1000 nails) and the task is completed.

If only two people or machines working together use this simpler formula:

Simplified Combined Work Formula:

Total Time = AB/A + B

Where A and B are the times it takes the two people or machines alone to finish the job. Here you are solving for the total time directly and not it’s reciprocal like above. The total time is the time both people or machines take to do the task when working together.

REMEMBER: This formulas only work if you are given the times and they are all for the same-sized task (e.g. making 1000 nails) and the task is completed. If both people or machines are working together but independently and their task is to finish more than one set of the task (e.g. 2 batches of 1000 nails when the times given were for one batch, or 2 rooms to paint when the times given were for painting one room) you have to remember to multiply your final result by 2 otherwise you would count as if they are both working on one batch or one room etc.

Another way to calculate the total time is to do this:

1. Le’t say you are given the two times for the two entities. Then you can quickly crete the rates for the two by taking the reciprocal of the times. (e.g. one entity finishes work in 3 hours, the other in 5. I.e. rates are 1/3 and 1/5)
2. Add the two rates to one combined rate. (1/3 + 1/5 = 8/15)
3. Use the Work/Rate formula to find the total time:

W = R x T i.e. T = W/R

(remember that W=1 in these because it’s one job being done)

Question 15

Questions With 2 Trains Moving Towards Each Other:

Answer 15

Use backsolving by looking at where the trains are at the time in the answer choice to check if they are at the same spot as that’s what the questions often ask for. Remember: When checking whether they are at the same spot remember to consider the entire distance, e.g. if entire distance is 90 miles then if train A is 70 miles away from its starting point and train B has traveled 20 miles from its starting point they are both at the same spot.

Question 16

Hour-by-Hour Approach to Combined-Work Problems:

Answer 16

An alternative to adding the times each person or machines needs to complete the work is to break down the work that each one does on an hour-by-hour basis to see how much (what fraction of the) work each person or machine does.

E.g. if one person does the work in 9 hours, then in one hour he has completed 1/9 of the job and in 2 1/4 hours:

2x1/9 + 1/4 x 1/9 = 2/9 + 1/36 = 1/4

This means that in 2 1/4 hours this person has completed 1/4 of the job at hand.

To see how long this person needs to do the entire work we only need to multiply the time, 2 1/4 by 4, because after 2 1/4 he has completed 1/4 of the job so in order to see how long it take him to complete the whole job we have to multiply 2 1/4 by 4.

9/4 x 4 = 9

and of course then you would see that the person would take 9 hours to complete the entire job.

Question 17

Multiple Rate Problems:

Answer 17

GMAT will sometimes give you problems involving multiple rates. Use the data to set up your equations for the multiple rates. You will have to set up multiple rates.

RT = D

Look for relationships between the equations you set up to get all the data you need.

Question 18

Relative Rate Problems:

Answer 18

In these two bodies are traveling at the same time.

3 Possible Scenarios for Relative Rate Problems:

1. The bodies move toward each other.
2. The bodies move away from each other.
3. The bodies move in the same direction on same path.

In these don’t set up multiple RT = D equations. Rather create a third RT = D equation for the rate at which the distance between the bodies change.

E.g.:

1. Add separate rates if move toward each other:
   Two people decrease the distance between themselves at rate of 5+6 = 11mph
2. Add rates if move away from each other:
   Two people increase distance between themselves at a rate of 30+45 = 75.
3. Two people move in same direction at different rates:
   Person X and Y decrease distance between themselves at rate 8-5 = 3mph

Then set up RT = D equation.

Question 19

Average Rate:

Answer 19

REMEMBER: If person moves same distance at two different rates you can’t just add the rates to find the average rate. Because if the person will spend more time traveling at the slower rate, the average rate will always be closer to the slower of the two rates.

Solve by:

1. Find total combined distance and total combined time of travel.
2. Use this formula to calculate:

Average speed: Total Distance/Total Time

Often the problem does not give you a distance because it doesn’t matter. If someone walks at a rate of 15 miles per hour then that’s all you need to know. Just pick your own number for the distance but pick smart numbers (check what are common multiples of the given numbers).

So here’s how to solve:

You have the two rates. You picked a distance (same distance for both trips if problem allows).

Then first find the total time by adding the individual times. You can find the individual times by using the d = rxt formula. Add the distances, add the times and then make the final calculation by using Average speed = Total distance/total time to find average rate.

E.g. Lucy walks to school at 4mph, back home at 6mph. What’s her average walking rate?

1. Pick 12 for distance as multiple of both.

Total distance: 24 miles

2. Find individual times and add to total time:

tx4 = 12 t = 3

tx6 = 12 t = 2

Total time: 5 hours

3. Calculate Average rate:

5 x r = 24

r = 24/5 = 4,8 mph

Question 20

Work Problems:

Answer 20

Are a type of rate problem.

Use formula:

W = r x t

and

R = w/t

where w is the number of jobs completed, i.e. number of items produced.

Many work problems require you to calculate rate. They often ask you how long it takes for someone to do a job.

You can solve by looking at what fraction of the work someone has done after what time and add the fractions until you get the whole job done. Or you can find the rate first using r = w/t. Then find the time using t = w/r. where w is the total job that needs to get done.

(Side note: when the amount of work to be done is 1 unit the time and rate will always be reciprocals of each other because: 1 = rxt)

Question 21

Working Together: Add Rates

Answer 21

When two or more workers are performing the task add their rates together.

Scenario:

If Lucas completes 1/3 of a task in an hour, and Serena 1/2 of task in one hour, they together complete in one hour:

1/3 + 1/2 = 5/6 of task.

If one worker is undoing work of the other subtract rates.

Scenario:

If hose1 fills tank at rate of 3 gallons per minute and hose2 drains it at rate of 1 gallon per minute, tank is being filled at rate:

3-1 = 2 gallons per minute.

In most GMAT work questions need these two equations only:

w = rxt and

r1+r2+…=rtotal

REMEMBER: When work involves completing one task, the unit to use is 1. For instance if task is to fill a tank then use 1 for work. If it is to fill 2/3 of tank, use 2/3 for work. And if someone completes one task in 4 hours then rate = 1/4.

REMEMBER: Once you have calculated the rates for all workers, add their rates to find the total rate at which they will complete the job together.