2006 AMC 10A Problem 1

P1-10~1 minPrint

Linear Equations & Word Problems Set up the equation/formula and compute; no special trick needed

Sandwiches at Joe's Fast Food cost $\$3$ each and sodas cost $\$2$ each. How many dollars will it cost to purchase $5$ sandwiches and $8$ sodas?

Show solution without answering

Key insight

Find the cost of the sandwiches and the cost of the sodas separately, then add the two amounts.

Solution

Work out each kind of food on its own line, then add.

Sandwiches: $5$ sandwiches at $\$3$ each cost $5 \times 3 = 15$ dollars.
Sodas: $8$ sodas at $\$2$ each cost $8 \times 2 = 16$ dollars.
Together: $15 + 16 = 31$ dollars.

The answer is $\boxed{\textbf{(A)}\ 31}$ .

Why this works

When you buy several kinds of things, the bill is (how many) times (price of one) for each kind, added up. Writing each kind on its own line makes it easy to check that every price goes with the right item.

Concise solution

A shorter write-up, the way contest solutions are usually written. It may take a different route from the solution above.

Five sandwiches at $\$3$ apiece come to $5 \cdot 3 = 15$ dollars. Eight sodas at $\$2$ apiece come to $8 \cdot 2 = 16$ dollars.

Adding the two subtotals gives $15 + 16 = 31$ dollars.

The answer is $\boxed{\textbf{(A)}\ 31}$ .

Review in the AoPS books

Introduction to Algebra, Chapter 3: One-Variable Linear Equations (§3.3, §3.4)
Introduction to Algebra, Chapter 7: Proportion (§7.4)