Question type

Border, frame and path area problems on the SAT

A border of width $x$ adds $x$ to both sides of each dimension, so an 8 by 10 rectangle becomes $(8 + 2 x)$ by $(10 + 2 x)$ . Set that product equal to the total area. If only the border's area is given, use outer area minus inner area. Keep the positive root.

These questions put a frame, deck, path or margin of uniform width around or inside a rectangle and ask for the width, or for the equation that finds it.

It appeared on two of the five official practice tests we analysed. The algebra is routine; the points are lost setting up the equation.

On 2 of 5 official practice tests
Desmos helps

Why students miss it

Adding $x$ once. A border on both sides adds $2 x$ to each dimension: $(8 + 2 x) (10 + 2 x)$ , not $(8 + x) (10 + x)$ .

Using the total area when only the border's area is given. Then the equation is outer area minus inner area, as in $(30 + 2 x) (40 + 2 x) - 1200 = 456$ .

Keeping both roots. A width cannot be negative.

How to spot it

"Uniform width" with a frame, border, deck, path or margin.
A rectangle's dimensions plus an area for the larger or smaller shape.
Answer choices that are equations, or that are lengths.

The method

Set up, then solve

Draw the rectangle and label the border $x$ on every side.

Border outside: each dimension grows by $2 x$ .
Margin inside: each dimension shrinks by $2 x$ .
Area of the border alone: outer area minus inner area.

With Desmos

Once the equation is right, Desmos solves it in one step.

Type the equation exactly as you set it up, for example (30+2x)(40+2x)-1200=456.
Desmos draws the solutions as vertical lines; the positive one is the width.

Desmos cannot check the setup. If you write $x$ where you need $2 x$ , it will solve the wrong equation correctly.

Two real questions

From the 1500prep bank, the way the app shows them. Answer first, then open the explanation.

1Question 1 of 2#706Adv

A rectangular photograph that measures 8 inches by 10 inches is surrounded by a frame of uniform width $x$ inches. The combined area of the photograph and the frame is 168 square inches. Which equation could be used to find the value of $x$ ?

Pick an answer, then check it.

Show the answer and explanation

The frame adds width

x

on both sides of each dimension, so the outer rectangle is

(8 + 2 x)

(10 + 2 x)

. Its area is the combined area, 168:

(8 + 2 x) (10 + 2 x) = 168

2Question 2 of 2#707Adv

A rectangular swimming pool measures 30 feet by 40 feet. A concrete deck of uniform width $x$ feet is built around the outside of the pool. If the area of the deck alone is 456 square feet, what is the value of $x$ ?

Pick an answer, then check it.

Show the answer and explanation

Deck area = outer area − pool area. Outer rectangle is

(30 + 2 x)

(40 + 2 x)

, so

(30 + 2 x) (40 + 2 x) - 1200 = 456

. Expanding:

4 x^{2} + 140 x = 456

, or

x^{2} + 35 x - 114 = 0

, which factors as

(x + 38) (x - 3) = 0

. So

x = 3

Questions students ask

Why is the frame 2x and not x?

The frame is on both sides of the picture, so each dimension grows by x on the left and x on the right, or top and bottom: 2x in total.

What if the question gives only the area of the border?

Write the outer area minus the inner area equals the border's area, then solve the resulting quadratic.

Practise this type until it is automatic

Five more border-and-frame questions, checked instantly, with every wrong choice explained. Free, no card.

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