Certain two-step equations, where \(x\) appears exactly once, as the innermost and leftmost operand, can be solved mentally. These are equations like

$$\begin{align} & (x \cdot a) + b = c \\ & (x - a) / b = c \\ & \ldots \end{align}$$Notice that to solve such equations, we start with \(c,\) the output, then repeatedly whittle down until only \(x\) remains. So to solve the equation

$$(x \cdot a) + b = c$$we'd first subtract \(b\)

$$x \cdot a = c - b$$Then we'd divide by \(a\)

$$x = (c - b) / a$$Thus, equations of the form

$$(x \cdot a) + b = c$$can be solved mentally exactly when \((c - b) / a\) can be evaluated mentally. Now you try. Derive a similar formula for equations of the form \((x + a) / b = c.\)

Here's the solution:

To solve the equation

$$(x + a) / b = c$$we'd first multiply by \(b\)

$$x + a = c \cdot b$$Then we'd subtract \(a\)

$$x = c \cdot b - a$$This strategy of repeatedly doing the inverse can be used to mentally solve both word problems and mathematical ones. Here are a few word problems for you to try:

I'm thinking of a number. If I double my number and add 6 the answer is 42. What's my number?

Solution: \((42 - 6) / 2 = 18\)

I'm thinking of a number. If I multiply my number by 6 and add 4 the answer is 70

Solution: \((70 - 4) / 6 = 11\)

I'm thinking of a number. If I add 3 to my number then divide the result by 2, the answer is 6.5. What's my number?

Solution: \((6.5 \cdot 2) - 3 = 10\)

For more problems like this, go here. Refresh that page for even more.

Next, give your students these challenges:

- Change Around by NRICH
- A Chain of Eight Polyhedra by NRICH

Conclude by leading this investigation:

Locker Room Prank

by MathPickle