Shirts cost $12 and hats cost $8. Ana can spend at most $100. Can… Step by Step

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The question

Question: Shirts cost $12 and hats cost $8.

Ana can spend at most $100.

Can she buy 6 shirts and 5 hats?

1 Find what we know

What do we know?

Shirts cost $12 each.

Hats cost $8 each.

Ana can spend $100 or less.

What do we want?

To know if 6 shirts and 5 hats fit.

2 Name the unknowns

We don't know how many shirts.

We call that number x.

x = number of shirts

We don't know how many hats either.

It is a different number, so it gets its own letter: y.

y = number of hats

3 Write the inequality

One shirt costs $12.

So x shirts cost 12 times x: 12x.

x shirts: 12 × x = 12x

One hat costs $8.

So y hats cost 8 times y: 8y.

y hats: 8 × y = 8y

At most $100 means $100 or less.

That is the sign less than or equal to.

at most $100 = $100 or less: ≤

In words: the shirt money plus the hat money is at most $100.

12x + 8y = 100

So the inequality is 12x + 8y ≤ 100.

12x + 8y ≤ 100

Careful!

At most means 100 or less.

So the sign is less than or equal to, not more than.

Careful: not 12x + 8y ≥ 100 ✗

4 Put the point in

6 shirts: x = 6.

5 hats: y = 5.

As a point, that is (6, 5).

x = 6 , y = 5 (6, 5)

We put them in.

Is it less than or equal to 100?

We don't know yet.

12 × 6 + 8 × 5 ≤? 100

12 times 6 is 72.

8 times 5 is 40.

12 × 6 = 72

8 × 5 = 40

72 plus 40 is 112.

72 + 40 ≤? 100

Is 112 less than or equal to 100?

No.

That is false.

112 > 100 false ✗

5 Answer the question

No: 6 shirts and 5 hats cost $112.

6 Check

Let's check with the money.

Ana would need $12 more.

So Ana can't.

It works!

$112 − $100 = $12 too much

Let's look back.

First: find what we know.

Then: name the unknowns.

Then: write the inequality.

Then: put the point in.

Then: answer the question.

Last: check.

Answer