Direct solving

This chapter applies to all AMC 8/10/12 test takers.

Some questions can be solved directly. These problems usually test whether you recognize the right equation (or set of operations) and can carry out the calculation accurately.

Not every direct-solving problem uses a named formula. Sometimes the problem simply describes what to do, and your job is to translate that into an equation and solve.

Many AMC problems combine multiple strategies, but in this chapter the focus is narrower: the questions are designed to practice routine calculation and direct solving. You should practice solving these without a calculator.

The simplest direct-solving questions don’t require you to identify a specific formula. They give you an equation and ask you to solve for a variable.

Example: The question below is from 2020 AMC 10A

What value of $x$ satisfies $x-\frac{3}{4}=\frac{5}{12}-\frac{1}{3}$?
A. $-\frac{2}{3}$
B. $\frac{7}{36}$
C. $\frac{2}{12}$
D. $\frac{2}{3}$
E. $\frac{5}{6}$

Slightly more challenging questions give you the inputs but require you to recognize which equation connects them. In the next problem, you need the arithmetic mean relationship.

Example: The question below is from 2020 AMC 10A

The numbers $3,5,7,a,$ and $b$ have an average (arithmetic mean) of $15$. What is the average of $a$ and $b$?
A. $0$
B. $15$
C. $30$
D. $45$
E. $60$

The most involved direct-solving questions require multiple steps. You typically find an intermediate value first, then use it in another calculation to reach the final answer. In the next problem, you can write $x$ in terms of $y$ and substitute into another equation.

Example: The question below is from 2003 AMC 10B

The first four terms in an arithmetic sequence are $x+y$, $x-y$, $xy$, and $\frac{x}{y}$, in that order. What is the fifth term?
A. $-\frac{15}{3}$
B. $0$
C. $-\frac{6}{5}$
D. $\frac{20}{27}$
E. $\frac{123}{40}$

Review the equations glossary to learn the key equations you’re expected to know for the AMC. Every equation in the glossary is introduced elsewhere in this course.

Common themes

• If you are told to round your final solution, make sure never to round the intermediate variables on the way to the solution. This could make your final answer less accurate.
• Before solving, look for opportunities to simplify. This can make the calculations much less complicated.
• If answer choices include $\pi$, the question will very likely require you to use an equation that includes $\pi$. Always look out for characteristics of answer choices that may indicate what equations to use.
• When answer choices involve variables or full sentences, the question is likely not a direct solving opportunity.