Times as Many: Comparison and Multi-Step Word Problems

Times as many means multiply

$$35 = 5 \times 7$$

That can be read as a comparison:

• $35$ is five times as many as $7$
• $35$ is seven times as many as $5$

Both are true. The equation holds two comparisons at once, one for each factor.

Why “times as many” is not “more than”

These two phrases sound similar and do opposite things.

Phrase	Operation	With $7$ and $5$
$5$ more than $7$	add	$12$
$5$ times as many as $7$	multiply	$35$

Same two numbers. Answers that are nowhere near each other.

Look for the word “times”. It is the signal to multiply. Without it, phrases like more than, fewer than and increased by all mean adding or subtracting.

Finding how many times as many

Multiplication going one way means division coming back.

Ben has $6$ marbles. Amy has $42$. How many times as many does Amy have?

$$42 \div 6 = 7$$

Seven times as many.

This is the missing factor again: $? \times 6 = 42$.

Question	Operation
How many is $5$ times as many as $8$?	$5 \times 8$
$40$ is how many times as many as $8$?	$40 \div 8$
$40$ is $5$ times as many as what?	$40 \div 5$

Two-step problems

Most real problems need more than one step. The method is to finish one step and write the answer down before starting the next.

A store has $4$ boxes with $6$ pens in each. It sells $9$ pens. How many are left?

Step 1 — how many pens to start:

$$4 \times 6 = 24$$

Step 2 — how many after selling:

$$24 - 9 = 15$$

Fifteen pens left.

Writing the $24$ down is the whole technique. Trying to hold it in your head while doing the next step is where most errors in these problems come from.

Writing it as one equation

The two steps can be joined into one statement, with a letter for the answer:

$$p = (4 \times 6) - 9$$

The parentheses keep the multiplication first, which the order of operations would do anyway — but writing them makes the intended order visible to a reader.

Checking the answer makes sense

An answer to a word problem describes something real, so it can be checked against the situation.

Warning sign	What it usually means
bigger than the total	added where you should have taken away
a negative amount	subtracted the wrong way round
not a whole number of objects	multiplied or divided the wrong pair
far larger than expected	multiplied where you should have added

In the pen problem, $15$ is less than the $24$ started with, which is right for a shop that sold some. An answer of $33$ would mean the store gained pens by selling them.

This check costs one sentence and catches the errors that matter.

Worked examples

Common mistakes

Practice problems