Before you read

Answer first — see what you already know.

A farmer has chickens and cows. There are 30 animals total and 80 legs. How many chickens are there? (Chickens have 2 legs, cows have 4.)

A
$10$
B
$15$
C
$20$
D
$25$

Check your answer

Answer:

Before you read

Answer first — see what you already know.

A cashier has $$5$ bills and $$10$ bills totaling $$175$ . There are 25 bills in all. How many $$10$ bills are there?

A
$5$
B
$10$
C
$15$
D
$20$

Check your answer

Answer:

Theory

Setting Up Systems from Word Problems

Most system word problems on the SAT follow a pattern:

Two unknowns $\to$ define two variables
Two relationships $\to$ write two equations
Solve $\to$ use substitution or elimination

Common types:
1. Quantity + Value: items and costs, tickets and prices
2. Mixture: combining solutions, blending ingredients
3. Distance/Rate: two objects moving
4. Age: relationships between people's ages
5. Comparison: one quantity related to another

Setting up a system from a word problem

Theory

The Quantity-Value Template

The most common SAT system word problem:

Equation 1 (Quantity): total number of items
$x + y = \text{total items}$

Equation 2 (Value): total value/cost
$a \cdot x + b \cdot y = \text{total value}$

where $a$ and $b$ are the per-item values.

Example 1

A theater sells adult tickets for $\$12$ and student tickets for $\$7$ . If 200 tickets were sold for a total of $\$1{,}900$ , how many of each type were sold?

Let $a$ = adult tickets, $s$ = student tickets.

Quantity: $a + s = 200$

Value: $12a + 7s = 1900$

From eq1: $s = 200 - a$ .

Substitute: $12a + 7(200 - a) = 1900$

$12a + 1400 - 7a = 1900$

$5a = 500$ $\to$ $a = 100$ .

$s = 200 - 100 = 100$ .

100 adult tickets and 100 student tickets

12(100) + 7(100) = 1200 + 700 = 1900

✓

Example 2

A store sells two types of coffee. Blend A costs $\$8$ /lb and Blend B costs $\$12$ /lb. A customer buys 5 lbs total for $\$48$ . How many pounds of Blend A?

Let $a$ = lbs of A, $b$ = lbs of B.

$a + b = 5$ and $8a + 12b = 48$ .

$a = 5 - b$ . Substitute: $8(5-b) + 12b = 48$ .

$40 - 8b + 12b = 48$ $\to$ $4b = 8$ $\to$ $b = 2$ .

$a = 5 - 2 = 3$ .

3 lbs of Blend A

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