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SAT MATH · DESMOS PART 1

By hand. With Desmos.

Use a calculator when the graph helps. Always interpret the result in the question’s context.

EXAMPLE 1 OF 2

Successive percentage changes

During March, the number of plants in a nursery increased by p%. During April, the number of plants decreased by (p − 5)% of the number present at the end of March. At the end of April, the nursery had 2% more plants than it had at the beginning of March. If p > 5, what is the value of p?

By hand

  1. Let N be the initial number of plants. After March, the number is N(1 + p/100).
  2. After April, the number is N(1 + p/100)(1 − (p − 5)/100). Set this equal to 1.02N and cancel N.
  3. Multiply by 10,000: (100 + p)(105 − p) = 10,200.
  4. Expand and rearrange: 10,500 + 5p − p² = 10,200, so p² − 5p − 300 = 0.
  5. Factor: (p − 20)(p + 15) = 0. Since p > 5, p = 20.
  6. Verify independently using an initial count of 100 plants: a 20% increase gives 120 plants, and a 15% decrease then gives 120 × 0.85 = 102 plants, exactly 2% above the initial count.

With Desmos

The graph shows why there are two algebraic candidates and why the context still matters.

Copy and paste each line into a separate expression:

y=(1+x/100)(1-(x-5)/100)
y=1.02

Set x from -25 to 30 and y from 0.9 to 1.1 so both crossings are visible.

The graphs meet at x = -15 and x = 20. The condition p > 5 leaves p = 20. Verify: 1.20 × 0.85 = 1.02.

Open the SAT Desmos calculator ↗
Why this approach works

A percentage change multiplies the current amount. April starts with March’s increased total, so the two factors multiply; adding or subtracting the percentages would use the wrong starting amount.

After canceling the initial count, the expression gives final amount divided by initial amount for each possible percentage x. A final ratio of 1.02 means 102% of the original amount. The horizontal line y = 1.02 therefore asks which percentages produce exactly that ratio.

At each crossing, both expressions have the same output, so that x solves the original equation. Desmos finds candidates; the condition p > 5 chooses the valid one. The x-coordinate is the percentage, while the y-coordinate is a ratio.

EXAMPLE 2 OF 2

Read the requested difference from a system

At a museum, two adult tickets and three student tickets cost $55. Four adult tickets and one student ticket cost $65. What is the difference, in dollars, between the adult and student ticket prices?

By hand

  1. Let a be the adult price and s the student price: 2a + 3s = 55 and 4a + s = 65.
  2. Multiply the second equation by 3, then subtract the first: 10a = 140, so a = 14.
  3. Then s = 65 - 4(14) = 9. The requested difference is 14 - 9 = 5 dollars.

With Desmos

The intersection provides both prices, which you must interpret before answering the question.

Copy and paste each line into a separate expression:

2x+3y=55
4x+y=65

Use x for adult price and y for student price. Set both axes from 0 to 20.

The intersection is (14, 9). It gives the two prices, not the requested answer. Their difference is 5 dollars.

Open the SAT Desmos calculator ↗
Why this approach works

Each equation describes every adult/student price pair that could produce one receipt total. We need a pair that makes both receipts true at once, so we look for a point lying on both lines.

An intersection shares one x-coordinate and one y-coordinate across both equations. With x defined as the adult price and y as the student price, (14, 9) means adult tickets cost $14 and student tickets cost $9. Those values satisfy both receipts: 28 + 27 = 55 and 56 + 9 = 65.

The question asks for a difference, so the intersection is an intermediate result. Subtract the two prices: 14 − 9 = 5. Naming the axes before graphing prevents a coordinate from being mistaken for the final answer.

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