From words to equations

From words to math

Everything in this course has been about solving equations someone handed you. In real problems you have to write the equation yourself. The trick is to play with the situation first and notice the pattern, then write the pattern down.

Build it before you write it

A gym charges a $20 sign-up fee plus $15 per month. After how many months will you have paid $110?

Add months one at a time and watch the total.

20

$20

target $110

$20 + 15 \cdot 0 = 20$$90 to go.

−0 months+

Every month adds another $15$ on top of the starting $20$. After $m$ months, that's $20 + 15m$. You want that to be $110$, so the equation is:

Clicking found the answer here, but it would take a long time for a total of $11,000. The equation is just as fast either way.

A routine that works

1. Name the unknown. Pick a letter and say what it stands for.
2. Translate. Turn the sentences into an equation or inequality. Trying a few numbers first helps you see the pattern.
3. Solve it.
4. Answer the question in words, and check that it makes sense.

Comparing two plans

Plan A costs $40 plus $5 per class. Plan B costs $10 per class with no fee. After how many classes do they cost the same?

Plan A starts ahead but grows slowly. Plan B starts at zero but grows fast. Watch Plan B catch up.

Plan A

40

$40

Plan B

$0

Plan A: $40 + 5 \cdot 0 = 40$Plan B: $10 \cdot 0 = 0$Plan B is cheaper after 0 class.

−0 class+

Let $c$ be the number of classes. "Cost the same" means the two expressions are equal:

Both plans cost $80 at 8 classes. After that, Plan A is the better deal.

Inequalities: "at most" and "at least"

You have $50. Movie tickets cost $12 each and you'll spend $8 on snacks. How many tickets can you buy?

8

$8

budget $50

$8 + 12 \cdot 0 = 8$$8 fits the $50 budget, with $42 to spare.

−0 tickets+

Let $t$ be the number of tickets. You can spend at most $50, so this is an inequality:

You can't buy half a ticket, so the answer is at most 3 tickets. Interpreting the answer matters: the math says $3.5$, but the situation says $3$.

Consecutive numbers

Three consecutive whole numbers add to $57$. What are they?

If the first is $n$, the next two are $n + 1$ and $n + 2$.

The numbers are $18$, $19$ and $20$. Sanity check: $57 \div 3 = 19$, the middle number.

Key ideas

• Play with the situation first: a few concrete numbers reveal the pattern.
• Name the unknown, then translate piece by piece: "is" is $=$, "at least" is $\geq$, "at most" is $\leq$.
• Answer the question that was asked, and make sure the answer fits the situation.