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Rates &
Averages

Master the formulas and strategies for solving rate problems and average (mean) problems on the ISEE Upper Level Quantitative Reasoning section.

ISEE Upper Level

What Is a Rate?

A rate is a ratio that compares two quantities measured in different units. It tells you how much of one thing occurs per unit of another.

Speed
\(\text{miles} \div \text{hours}\) gives miles per hour (mph)
Pay Rate
\(\text{dollars} \div \text{hours}\) gives dollars per hour
Flow Rate
\(\text{gallons} \div \text{minutes}\) gives gallons per minute

What Is an Average?

The average (arithmetic mean) is the value you get when you distribute the total sum evenly across all items. It represents the center of a data set.

This simple formula can be rearranged to find any one of the three quantities: average, sum, or count, when the other two are known.

The Rate Triangle

The three quantities in a rate problem: Distance, Rate, and Time, are connected by a single relationship. Cover the one you need to find.

\(D\)
\(R\)
\(T\)
\(\times\)
\(D = R \times T \quad R = \frac{D}{T} \quad T = \frac{D}{R}\)

The Average Triangle

Averages follow the exact same structure as rates. The three quantities: Sum, Average, and Count, form their own triangle.

\(S\)
\(A\)
\(N\)
\(\times\)
\(S = A \times N \quad A = \frac{S}{N} \quad N = \frac{S}{A}\)

Rates vs Averages

Though they use different vocabulary, rates and averages share the same underlying structure: a total divided by a count.

Rate Problems Average Problems
Total Distance (or amount) Sum of values
Per-unit value Speed (rate) Mean (average)
Count Time Number of items
Key formula \(D = R \times T\) \(S = A \times N\)
Common trap Averaging two speeds Forgetting to use sum

How to Solve Rate & Average Problems

1
Identify the type: is it a rate problem (distance/speed/time) or an average problem (sum/mean/count)?
2
Write the formula and identify which two quantities are given and which is unknown.
3
Convert units if needed: make sure distance, rate, and time (or sum, average, count) all use compatible units.
4
Solve algebraically: plug in, rearrange, and compute. For multi-leg trips, find each total separately then combine.
5
Check reasonableness: does your answer make sense in context? Is the average between the extremes?

Example 1: Basic Rate

A car travels 240 miles in 4 hours. What is its average speed?

Start End 240 miles 4 hours

Example 2: Finding the Sum

The average of 5 test scores is 88. What is the sum of all five scores?

Follow-up: If a 6th score of 94 is added, the new average is:
\(\dfrac{440 + 94}{6} = \dfrac{534}{6} = \) 89

Example 3: Average Speed Trap

A runner goes 3 miles at 6 mph, then 3 miles at 3 mph. What is the average speed for the whole trip?

3 mi @ 6 mph
\(T = 0.5\) hr
 3 mi @ 3 mph
\(T = 1\) hr
 Total: 6 mi in 1.5 hr

Example 4: Weighted Average

A class of 20 students scored an average of 75. Another class of 30 students scored an average of 85. What is the combined average?

1
Sum for class 1: \(20 \times 75 = \) 1500
2
Sum for class 2: \(30 \times 85 = \) 2550
3
Combined average: \(\dfrac{1500 + 2550}{20 + 30} = \dfrac{4050}{50} = \) 81

Example 5: Combined Work Rate

Pipe A fills a tank in 6 hours. Pipe B fills it in 3 hours. How long to fill it together?

1
Rate of A: \(\dfrac{1}{6}\) tank per hour
2
Rate of B: \(\dfrac{1}{3}\) tank per hour
3
Combined rate: \(\dfrac{1}{6} + \dfrac{1}{3} = \dfrac{1}{6} + \dfrac{2}{6} = \dfrac{3}{6} = \dfrac{1}{2}\) tank per hour
4
Time: \(\dfrac{1}{1/2} = \) 2 hours

Why You Can't Average Rates

When two legs of a trip cover the same distance at different speeds, the average speed is the harmonic mean, not the arithmetic mean.

This shortcut only works when the distances are equal. When distances differ, always use total distance ÷ total time.

Finding a Missing Value

A common ISEE pattern: "The average of 4 numbers is 15. Three of the numbers are 12, 18, and 10. Find the fourth."

1
Find the total sum: \(\text{Sum} = 15 \times 4 = \) 60
2
Sum the known values: \(12 + 18 + 10 = \) 40
3
Subtract: \(60 - 40 = \) 20

Common Mistakes

✕ Averaging Speeds
Adding two speeds and dividing by 2 only works if the time is equal, not the distance. Use total distance ÷ total time.
✕ Forgetting Units
Mixing hours with minutes or miles with kilometers leads to wrong answers. Always convert to the same units first.
✕ Ignoring Weights
When groups have different sizes, you cannot simply average the averages. You must use the weighted formula.

Formula Summary

Situation Formula Key Tip
Basic rate \(R = \dfrac{D}{T}\) D, R, T triangle
Average (mean) \(A = \dfrac{S}{N}\) Find sum first
Weighted average \(\dfrac{S_1 + S_2}{N_1 + N_2}\) Don't average averages
Combined work \(\dfrac{1}{T} = \dfrac{1}{T_1} + \dfrac{1}{T_2}\) Add rates, not times
Harmonic mean \(\dfrac{2ab}{a+b}\) Equal-distance trips only

Question 1 of 5

A train travels 360 miles in 5 hours. What is its average speed?

Question 2 of 5

The average of 6 numbers is 14. If one number is removed, the average of the remaining 5 numbers is 12. What number was removed?

Question 3 of 5

A cyclist rides 20 miles at 10 mph, then 20 miles at 20 mph. What is the average speed for the entire trip?

Question 4 of 5

Machine A completes a job in 4 hours. Machine B completes the same job in 12 hours. Working together, how long will they take?

Question 5 of 5

A group of 10 students has an average score of 80. A second group of 15 students has an average score of 90. What is the combined average?