# Midpoint Elasticity Practice: Revenue, Slope, and Missing Quantities

*2026-10-30*

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.

![A textile worker guides yarn between a full wooden reel and a nearly empty bobbin](https://czaixyl0esghkd4s.public.blob.vercel-storage.com/marketing/69a0b2d19c9338de57482d70948c7c849a45020b426911ade80f006931bcd054.webp)

## 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](https://openstax.org/books/principles-microeconomics-3e/pages/5-1-price-elasticity-of-demand-and-price-elasticity-of-supply).

Let `P1, Q1` be the first price and quantity, and `P2, Q2` the second:

```text
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](/blog/percentage-points-vs-percent-change/) 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**.

```text
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](https://openstax.org/books/principles-microeconomics-3e/pages/5-3-elasticity-and-pricing).

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:

1. The average price and average quantity.
2. Both proportional changes and the signed elasticity.
3. 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:

```text
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`.

```text
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:

```text
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**.

```text
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:

```text
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](https://ocw.mit.edu/courses/15-024-applied-economics-for-managers-summer-2004/4e3362012c473cdb2445bc69ddf97591_lec3.pdf) 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`:

```text
(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**.

```text
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:

```text
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](/blog/how-often-should-you-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.*

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