Midpoint Elasticity Practice: Revenue, Slope, and Missing Quantities
An elasticity of 1.00 on your calculator can still describe inelastic demand. In one exercise below, price rises, quantity falls, and revenue increases by just 8. The exact fraction explains what the rounded answer hides.
This midpoint elasticity practice set starts with a worked example, followed by seven problems and full solutions. You'll reverse a calculation, find a missing quantity, and compare three intervals on the same straight demand line. Keep a sheet of paper nearby: the useful part is choosing the denominator yourself.
Use the average of each pair
Midpoint price elasticity of demand divides the proportional quantity change by the proportional price change. Both use the average of their endpoints as the base. This measures elasticity over an interval, also called arc elasticity. OpenStax gives the midpoint formula and classifications.
Let P1, Q1 be the first price and quantity, and P2, Q2 the second:
Average price = (P1 + P2) / 2
Average quantity = (Q1 + Q2) / 2
Signed elasticity = [(Q2 − Q1) / average quantity]
÷ [(P2 − P1) / average price]
E = absolute value of the signed elasticity
You can multiply both proportional changes by 100 to express them as percentages; the factors cancel in the ratio. The averages are calculation bases, not observations from another period.
For downward-sloping demand, the signed quotient is negative. Classify using its magnitude, E:
| Magnitude | Classification |
|---|---|
E < 1 |
Inelastic |
E = 1 |
Unit elastic |
E > 1 |
Elastic |
Ordinary percent change uses the starting value. That's still correct when the question asks for it; the percentage points vs percent change worksheet practices that calculation.
What the fictional data assume
Except in exercise 7, treat each pair as two points on the same downward-sloping demand curve, with other demand determinants held fixed. Prices and quantities are positive. Quantities refer to sales periods of equal length, units stay consistent, and everyone pays the same price at each endpoint.
The result describes the interval you calculate. It doesn't establish how demand responds elsewhere on the curve.
Work through one pair first
Price rises from 8 to 12, and quantity demanded falls from 150 to 90.
Average price = (8 + 12) / 2 = 10
Average quantity = (150 + 90) / 2 = 120
Price change = (12 − 8) / 10 = 2/5 = 40%
Quantity change = (90 − 150) / 120 = −1/2 = −50%
Signed elasticity = (−1/2) / (2/5) = −5/4
E = 5/4 = 1.25 → elastic
The quantity change is proportionally larger than the price change on these average bases.
Using starting values would give (−60/150) / (4/8) = −4/5. Reversing that calculation gives (60/90) / (−4/12) = −2. With midpoint bases, reversing the endpoints changes both signs and preserves the quotient. You'll check that in exercise 2.
Check revenue separately
Total revenue is P × Q. Here it falls from 8 × 150 = 1,200 to 12 × 90 = 1,080, a decrease of 120.
For a price rise under our demand assumptions, midpoint E < 1 means endpoint revenue rises, E > 1 means it falls, and E = 1 means it is equal. A price fall reverses those directions. OpenStax explains the elasticity and revenue relationship.
Use the multiplication as a check on your classification. Revenue excludes costs, so this calculation doesn't tell you profit or settle a real pricing decision.
Seven price elasticity of demand practice problems
Stop before the answer section and try the problems yourself. For exercises 1–3 and 6, show:
- The average price and average quantity.
- Both proportional changes and the signed elasticity.
- The magnitude, classification, and revenue at each endpoint.
Keep exact fractions until you've classified the interval. Exercises 4, 5, and 7 have extra instructions.
1. A price rise with a smaller quantity response
Price rises from 6 to 9. Quantity demanded falls from 100 to 70.
2. Reverse the worked example
Price falls from 12 to 8. Quantity demanded rises from 90 to 150. Compare the elasticity with the worked example. Does revenue move in the same direction?
3. Test exact unit elasticity
Price rises from 9 to 15. Quantity demanded falls from 100 to 60.
4. One straight demand line, three intervals
Use the demand equation:
Q = 240 − 10P
Find quantity at prices 4, 8, 12, and 16. Calculate midpoint elasticity for each adjacent interval: 4 → 8, 8 → 12, and 12 → 16. Classify each interval and check revenue at all four points.
Then calculate ΔQ/ΔP, where Δ means change. On a graph with price on the vertical axis and quantity on the horizontal axis, what is the line's slope? Does a fixed slope give a fixed elasticity?
5. Find the missing quantity
Price rises from 18 to 22. Initial quantity is 150, and the midpoint elasticity magnitude is 2. Find the final quantity, assuming downward-sloping demand. Then calculate revenue at both endpoints.
The average quantity contains your unknown. Write it out before solving.
6. A rounded answer near one
Price rises from 10 to 12. Quantity demanded falls from 10,000 to 8,334. Your calculator displays the elasticity magnitude as 1.00 to two decimal places.
Is the interval exactly unit elastic? Use an exact fraction to classify it, and calculate the revenue difference.
7. Quantity changes while price stays fixed
Two separate observed periods report price 10 in both periods. Quantity falls from 100 to 80. You have no assurance that the observations lie on the same demand curve.
Can you calculate midpoint price elasticity from these records? Do they establish zero or infinite elasticity? Calculate revenue too, and separate what the arithmetic tells you from what the observations leave unknown.
Answers with the calculations shown
1. Inelastic, with revenue up 30
Average price is (6 + 9)/2 = 7.5; average quantity is (100 + 70)/2 = 85.
Price change = 3 / 7.5 = 2/5
Quantity change = −30 / 85 = −6/17
Signed elasticity = (−6/17) / (2/5) = −15/17
E = 15/17 ≈ 0.88235 → inelastic
Revenue rises from 6 × 100 = 600 to 9 × 70 = 630, a gain of 30. This agrees with the price rise over an inelastic interval.
2. Elastic again, with revenue up 120
Average price is still 10, and average quantity is still 120. Both changes reverse sign:
Price change = (8 − 12) / 10 = −2/5
Quantity change = (150 − 90) / 120 = 1/2
Signed elasticity = (1/2) / (−2/5) = −5/4
E = 5/4 = 1.25 → elastic
Revenue rises from 12 × 90 = 1,080 to 8 × 150 = 1,200, a gain of 120. The worked example covered the same interval in the other direction and lost 120. The elasticity is unchanged; the direction of the revenue change reverses.
3. Exactly unit elastic, with equal revenue
Average price is 12; average quantity is 80.
Price change = (15 − 9) / 12 = 1/2
Quantity change = (60 − 100) / 80 = −1/2
Signed elasticity = (−1/2) / (1/2) = −1
E = 1 → unit elastic
Revenue is 9 × 100 = 900 before and 15 × 60 = 900 after. The exact quotient equals one, and the endpoint revenues match.
4. The slope stays fixed while elasticity changes
Substitute each price into Q = 240 − 10P:
| Price | Quantity | Total revenue |
|---|---|---|
| 4 | 200 | 800 |
| 8 | 160 | 1,280 |
| 12 | 120 | 1,440 |
| 16 | 80 | 1,280 |
Every adjacent step adds 4 to price and subtracts 40 from quantity. The average bases change:
| Price interval | Average P | Average Q | Signed midpoint calculation | Magnitude and classification |
|---|---|---|---|---|
| 4 → 8 | 6 | 180 | (−40/180) / (4/6) = −1/3 |
1/3 ≈ 0.33333, inelastic |
| 8 → 12 | 10 | 140 | (−40/140) / (4/10) = −5/7 |
5/7 ≈ 0.71429, inelastic |
| 12 → 16 | 14 | 100 | (−40/100) / (4/14) = −7/5 |
7/5 = 1.4, elastic |
Revenue rises over the first two intervals, then falls over the third. Each result agrees with the corresponding elasticity classification.
The equation gives ΔQ/ΔP = −40/4 = −10 quantity units per currency unit. On the specified graph, slope is vertical change divided by horizontal change: ΔP/ΔQ = 4/(−40) = −1/10 currency unit per quantity unit. Name the axes before calling either ratio “the slope.”
Rearranging the midpoint formula shows why elasticity changes:
Signed arc elasticity = (ΔQ/ΔP) × (average P / average Q)
The first factor stays fixed on this line. The second changes from 6/180 to 10/140 to 14/100. Equal changes in units therefore produce different proportional responses.
Elasticity has no units; slope carries units. MIT's lecture on elasticity discusses how elasticity varies along a straight demand line. Our table calculates three intervals, rather than elasticity at individual points. Arc and point measures aren't interchangeable for every demand curve.
5. Final quantity is 100
Average price is (18 + 22)/2 = 20, so proportional price change is 4/20 = 1/5. Downward-sloping demand makes the signed elasticity −2. The required proportional quantity change is therefore −2 × 1/5 = −2/5.
Let the unknown final quantity be Q2:
(Q2 − 150) / [(Q2 + 150)/2] = −2/5
2(Q2 − 150) / (Q2 + 150) = −2/5
10(Q2 − 150) = −2(Q2 + 150)
10Q2 − 1,500 = −2Q2 − 300
12Q2 = 1,200
Q2 = 100
Check by putting 100 back into the formula. Average quantity is 125, so quantity change is −50/125 = −2/5. Dividing by price change 1/5 gives −2, as required.
A tempting shortcut is to reduce the initial 150 by 40%, giving 90. That uses a starting-value denominator. The midpoint equation requires the average of the initial and final quantities, so you have to solve for the final quantity first.
Revenue falls from 18 × 150 = 2,700 to 22 × 100 = 2,200, a decrease of 500.
6. Strictly inelastic, despite the 1.00 display
Average price is 11; average quantity is 9,167.
Price change = 2/11
Quantity change = −1,666/9,167
Signed elasticity = (−1,666/9,167) / (2/11) = −9,163/9,167
E = 9,163/9,167 ≈ 0.99956365
The numerator is smaller than the denominator, so E < 1. The interval is inelastic. Rounding to 1.00 hides the difference; it doesn't make the exact fraction equal to one.
Revenue confirms the small increase:
Before: 10 × 10,000 = 100,000
After: 12 × 8,334 = 100,008
Change: +8
Classify before rounding your reported answer. If you keep only the two-decimal display, you've discarded the information needed to distinguish this case from exact unit elasticity.
7. Undefined, with the demand curve unknown
Average price is 10, so proportional price change is 0. Average quantity is 90, giving a proportional quantity change of −20/90 = −2/9.
The quotient would be (−2/9) / 0, which is undefined. It isn't zero. These records also don't establish a perfectly elastic demand curve or an infinite elasticity value. They tell you that quantities differed at the same recorded price. A demand shift or other causes could explain that difference; the records don't identify the cause.
Revenue is still calculable: 10 × 100 = 1,000 and 10 × 80 = 800, a decrease of 200. Knowing those totals doesn't supply the missing evidence about a price response.
Keep one calculation choice for later
Once you've checked your answers, choose a mistake worth revisiting. Use a short recall prompt for the decision that went wrong, then retry a full problem without its solution in view.
| Front | Back |
|---|---|
| For midpoint elasticity between P = 8 → 12 and Q = 150 → 90, what are the two calculation bases? | Average price 10 and average quantity 120. |
| A demand interval has exact midpoint magnitude 9,163/9,167, displayed as 1.00. How is it classified? | Inelastic: the exact fraction is below 1. Classify before rounding. |
| Two records show the same price but different quantities. What happens to the midpoint price-elasticity calculation? | Price change is zero, so the quotient is undefined. The records alone don't identify the demand curve or the cause of the change. |
Attempt each answer before looking at the back, and revisit it later. The guide to how often to review flashcards explains how to space those attempts. Keep doing complete calculations too, especially the algebra in the missing-quantity problem.
Production note: AI tools substantially assisted with research, writing, and exercise construction. All price and quantity data are fictional. The worked example and answer arithmetic were checked with exact rational arithmetic in Python 3.14.7 on macOS 26.6.1. These checks are not market measurements, classroom trials, or learner-outcome tests. The illustration is AI-generated.