Multi-Step Problems: Comparisons and Unknowns | Grade 5 Math | LevelKite

You already know how to solve a multi-step problem. That is a problem that needs more than one operation. The operations are adding, subtracting, multiplying and dividing. In this lesson, the problems compare two amounts.

There are two common ways to compare. They sound alike, but they mean different things.

More than is about extra ones. "4 more than 6" means 6 with 4 extra ones added on.
Times as many is about equal groups. "4 times as many as 6" means 4 equal groups with 6 in each group.

The table shows both comparisons side by side. Each row gives the words, what they mean, and a number sentence, which is a line of math such as $6 + 4 = 10$ . With these numbers, the two results are far apart.

Two comparisons with the same numbers

Words	What the words mean	Number sentence
4 more than 6	6 with 4 extra ones	$6 + 4 = 10$
4 times as many as 6	4 equal groups of 6, which is $6 + 6 + 6 + 6$	$4 \times 6 = 24$
2 more than 30	30 with 2 extra ones	$30 + 2 = 32$
2 times as many as 30	2 equal groups of 30, which is $30 + 30$	$2 \times 30 = 60$

Worked example

Two comparisons in one problem

Ana has 48 stickers. Ben has 3 times as many stickers as Ana. Cal has 3 more stickers than Ana. How many stickers do Ben and Cal have together?

Find Ben's stickers. "3 times as many as Ana" means 3 equal groups of Ana's 48, so multiply. 48 is 40 and 8, so 3 groups of 48 are 3 groups of 40 plus 3 groups of 8.
$3 \times 48 = (3 \times 40) + (3 \times 8) = 120 + 24 = 144$
Find Cal's stickers. "3 more than Ana" means Ana's 48 with 3 extra ones, so add.
$48 + 3 = 51$
The question asks for Ben and Cal together, so add their two amounts.
$144 + 51 = 195$

Answer

Ben and Cal have 195 stickers together.

You solved the sticker problem one piece at a time. You already know another way: write the whole story in one line as an equation, a number sentence with an equals sign. The equals sign tells you that both sides are the same amount.

Now you will add one new thing to your equations: a letter. The unknown is a number you do not know yet. Here it is the number the question asks for. A letter holds the place of the unknown in the equation. You can pick any letter. It helps to pick one that reminds you of the story, such as $s$ for stickers.

Let $s$ stand for the number of stickers Ben and Cal have together. The table shows how each part of the story becomes part of the equation.

From the sticker story to an equation

Part of the story	Part of the equation
Ben has 3 times as many stickers as Ana, who has 48	$(3 \times 48)$
Cal has 3 more stickers than Ana, who has 48	$(48 + 3)$
Ben and Cal together	$+$ joins the two parts
The number you want to find	$s$

s = (3 \times 48) + (48 + 3)

Parentheses are the curved marks ( ). They group one amount that you work out first. Here each set of parentheses holds one person's stickers. You can read the equation before you work anything out: $s$ is 3 times 48, plus 3 more than 48.

Now work through it one operation at a time. Each new line changes just one part.

$s = 144 + (48 + 3)$ , because $3 \times 48 = 144$
$s = 144 + 51$ , because $48 + 3 = 51$
$s = 195$ , because $144 + 51 = 195$

This is the same answer you found before. The equation tells the whole story in one line, and the letter marks the number you are looking for.

Key concept

Read the comparison, then write the equation

Each comparison has its own meaning.

"5 more than an amount" means that amount with 5 extra ones. When you know the amount, you add 5.
"5 times as many as an amount" means 5 equal groups, each the size of that amount. When you know the amount, you multiply it by 5.

Write the whole story as one equation, with a letter for the unknown. Then work through the equation one operation at a time until you know the number the letter stands for.

Worked example

The letter stands alone

Grade 3 collected 245 bottle caps. Grade 4 collected 6 times as many bottle caps as grade 3. Grade 5 collected 180 more bottle caps than grade 4. How many bottle caps, $n$ , did grade 5 collect?

Write the equation. Let the letter n stand for grade 5's bottle caps. Grade 4 has 6 times as many as grade 3, so grade 4 has 6 equal groups of 245. Grade 5 has 180 more than grade 4, so add 180 extra ones to grade 4's amount. You can read the equation as: n is 180 more than 6 times 245.
$n = (6 \times 245) + 180$
Work out the parentheses first, because they hold one amount: grade 4's bottle caps. 245 is 200 + 40 + 5, so multiply each part by 6 and add the results.
$6 \times 245 = (6 \times 200) + (6 \times 40) + (6 \times 5) = 1{,}200 + 240 + 30 = 1{,}470$
Write the equation again with 1,470 in place of the parentheses. It still says the same thing, because 6 times 245 and 1,470 are the same amount.
$n = 1{,}470 + 180$
One operation is left. Add the 180 extra bottle caps to grade 4's 1,470. The letter stands alone, so this last number is the number it stands for.
$n = 1{,}470 + 180 = 1{,}650$

Answer

Grade 5 collected 1,650 bottle caps, so $n = 1{,}650$ .

In the bottle cap problem, the letter stood alone on one side of the equals sign. You worked out the other side, and the last number you got was the number the letter stands for.

Sometimes the unknown is one part of a total. Then the letter sits inside the equation with the other parts, like this: part + part + letter = total. You still go one operation at a time.

Work out each part that has only numbers in it.
Add the parts you know.
Now the equation says that a known amount plus the letter equals the total. A missing part is the total minus the part you know, so subtract.

Worked example

The letter is one part of a total

A school library has 2,000 books in all. Each one is a comic book, a story book or a fact book. The library has 125 comic books. It has 12 times as many story books as comic books. How many fact books, $f$ , does the library have?

Write the equation. The three kinds of books add up to the total of 2,000. Comic books are 125. Story books are 12 equal groups of 125. Fact books are the unknown, so the letter f holds their place.
$125 + (12 \times 125) + f = 2{,}000$
Work out the parentheses first. 12 is 10 + 2, so 12 groups of 125 are 10 groups plus 2 groups. 10 times 125 is 1,250, and 2 times 125 is 250. Add them: 1,250 + 250 = 1,500 story books.
$125 + 1{,}500 + f = 2{,}000$
Add the two parts you know, the comic books and the story books: 125 + 1,500 = 1,625.
$1{,}625 + f = 2{,}000$
Now 1,625 plus the letter f equals the total. A missing part is the total minus the part you know, so subtract 1,625 from 2,000.
$f = 2{,}000 - 1{,}625 = 375$
Check by putting 375 in place of f in the first equation. Both sides come to 2,000, so 375 fits the equation. This check cannot tell you whether the equation fits the story, so read the story once more as well.
$125 + (12 \times 125) + 375 = 125 + 1{,}500 + 375 = 2{,}000$

Answer

The library has 375 fact books, so $f = 375$ .

Here is the idea to keep. More than means extra ones. Times as many means equal groups. Decide which one a problem uses before you choose an operation. Then write the whole story as one equation with a letter for the unknown, and work through it one operation at a time.

Next, you will estimate the answer to a multi-step problem and use your estimate to decide whether a computed answer makes sense. You will also find and fix mistakes in solved problems.