Chapter 3 · Ratios and proportional relationships

Solving Proportional Word Problems and Finding Errors

Turn a proportional word problem into an equation, solve it, and check the answer. Then find and fix the first wrong line in someone else's work.

19 min read

You already know how to write an equation $y = kx$ from a table or from a graph. A word problem gives you the same information, only in sentences.

In this lesson you will pull the quantities out of a story, write the equation, and solve it, which means finding the missing value that makes the equation true. Then you will practice error analysis: reading finished work line by line to find the first place it goes wrong, explain why, and fix it.

Start with what proportional means. Two quantities $x$ and $y$ are in a proportional relationship when $\frac{y}{x}$ equals one constant $k$ , a number that stays the same, for every pair of values with $x \neq 0$ . You leave out $x = 0$ because you cannot divide by $0$ . Another way to say the same thing is $y = kx$ . Its graph is a straight line, and because $x = 0$ gives $y = 0$ , the line passes through $(0, 0)$ .

The number $k$ is the constant of proportionality. It is also the unit rate: the amount of $y$ that goes with $1$ unit of $x$ . That is why the point $(1, k)$ shows it. In a story you can say $k$ in words, such as 24 pages per minute.

Every story in this lesson tells you the relationship is proportional, and its quantities are amounts like minutes or pounds, so they are never negative. Use $y = kx$ only when the relationship is proportional.

A proportional word problem like this one is a small table in disguise.

A printer prints at a steady rate, so the number of pages it prints is proportional to the time it runs. It prints $96$ pages in $4$ minutes. How many pages does it print in $7$ minutes?

The story names two quantities: time in minutes and pages printed. Because the pages are proportional to the time, call the time $x$ and the pages $y$ , so that $y = kx$ . The story gives one complete pair, $4$ minutes with $96$ pages, and half of a second pair, $7$ minutes with an unknown number of pages. Sorted out, it looks like the tables you have already used.

The printer story sorted into pairs

Quantity	Complete pair	Pair to finish
Time in minutes, $x$	$4$	$7$
Pages printed, $y$	$96$	unknown

Key concept

One complete pair gives you the whole equation

When a relationship is proportional, $\frac{y}{x}$ is the same constant $k$ for every pair with $x \neq 0$ . So one complete pair with $x \neq 0$ is enough to find $k$ , and then $y = kx$ fits every pair, including the one the question asks about.

That gives you a plan: name $x$ and $y$ with their units, find $k$ from the complete pair, write $y = kx$ , put the value you know in place of its own letter, and solve for the other value.

k = \frac{y}{x} \text{ for } x \neq 0, \qquad y = kx

Worked example

Find the pages from the minutes

A printer prints at a steady rate, so the number of pages it prints is proportional to the time it runs. It prints $96$ pages in $4$ minutes. How many pages does it print in $7$ minutes?

Define the variables, the letters that stand for the two quantities, and give each one a unit. Let x be the time in minutes and y be the number of pages printed. The complete pair is x = 4, y = 96. The question gives x = 7 and asks for y.
Find k. The relationship is proportional, so y divided by x is the same constant k for every pair where x is not 0. The complete pair has x = 4, which is not 0, so you can divide. Pages divided by minutes gives pages per minute.
$k = \frac{y}{x} = \frac{96}{4} = 24$
Write the equation y = kx with k = 24. It says the number of pages equals 24 pages per minute times the number of minutes. The unit rate is 24 pages for 1 minute, which is the pair (1, 24).
$y = 24x$
Substitute 7 for x, which means replace the letter x with the number 7. This is allowed because the equation holds for every pair, and 7 is a number of minutes. Then multiply, splitting 24 into 20 and 4.
$y = 24 \times 7 = 20 \times 7 + 4 \times 7 = 140 + 28 = 168$
Check the new pair x = 7, y = 168. In a proportional relationship every pair where x is not 0 gives the same k, so 168 divided by 7 must be 24. It is.
$\frac{168}{7} = 24$

Answer

The printer prints $168$ pages in $7$ minutes.

Worked example

Find the pounds from the cost

Dried mango is sold by weight, and the cost is proportional to the weight. A bag holding $3$ pounds costs $12.75$ dollars. How many pounds can you buy with $34$ dollars?

Define the variables with units. The cost is proportional to the weight, so let x be the weight in pounds and y be the cost in dollars. The complete pair is x = 3, y = 12.75. This time the question gives a y-value, the cost of 34 dollars, and asks for the weight x.
Find k from the complete pair. Its x-value is 3, which is not 0, so divide y by x: 12 divided by 3 is 4, and 0.75 divided by 3 is 0.25. Dollars divided by pounds gives dollars per pound, so each pound costs 4.25 dollars.
$k = \frac{12.75}{3} = 4.25$
Write the equation y = kx with k = 4.25. Then substitute 34 for y, because 34 dollars is a cost and y stands for cost. Now x is the unknown.
$y = 4.25x \quad \text{so} \quad 34 = 4.25x$
Solve for x. The equation says 4.25 times x equals 34, so divide both sides by 4.25 to undo the multiplying. Dividing both sides by the same number keeps them equal as long as that number is not 0. Multiplying the top and bottom by 100 clears the decimal without changing the value, and 425 times 8 is 3400.
$x = \frac{34}{4.25} = \frac{3400}{425} = 8$
Check the new pair x = 8, y = 34. It gives the same k as the first pair, as every pair where x is not 0 must in a proportional relationship. A weight of 8 pounds is positive, so it makes sense in the story.
$\frac{34}{8} = 4.25$

Answer

You can buy $8$ pounds of dried mango with $34$ dollars.

Now use the same ideas to check work. Error analysis is more than redoing the problem. You read the work one line at a time and ask whether each line follows from the story and from the lines before it. The first line that fails is the error, and later lines may carry it forward.

Along with checking the arithmetic, ask three questions about a proportional word problem:

Units: Is each number matched with the right letter? In these stories $x$ and $y$ have different units, so a number's unit tells you whether it is an $x$ -value or a $y$ -value.
Order: Is $k$ found as $y \div x$ , using a pair with $x \neq 0$ , and is the equation $y = kx$ ?
Same constant: Does the final pair give the same $\frac{y}{x}$ as the pair in the story? When the relationship is proportional, every pair with $x \neq 0$ must.

The upside-down constant. A common mistake is to divide the complete pair in the wrong order. In the printer story that gives $k = \frac{4}{96} = \frac{1}{24}$ and the equation $y = \frac{1}{24}x$ . For $7$ minutes it predicts $\frac{7}{24}$ of a page, even though the printer made $96$ pages in only $4$ minutes.

Why it fails: $4 \div 96$ is minutes divided by pages, so $\frac{1}{24}$ means minutes per page. In $y = kx$ , with $x$ in minutes and $y$ in pages, $k$ must be $\frac{y}{x}$ from a pair with $x \neq 0$ , which is pages per minute. Only then does $k$ times a number of minutes give a number of pages. The value $\frac{1}{24}$ fits a different equation, $x = \frac{1}{24}y$ . The order question catches this error.

Here is a full piece of work to analyze. The problem: a hiker walks at a steady pace, so her distance is proportional to her time. She walks $9$ kilometers in $2$ hours. How long does she take to walk $27$ kilometers?

Dev's work is in the table below. Before you read the solution, test each line with the three questions and decide which line is the first wrong one.

Dev's work on the hiker problem

Line	What Dev wrote
1	Let $x$ be the time in hours and $y$ be the distance in kilometers.
2	$k = \frac{9}{2} = 4.5$ kilometers per hour
3	$y = 4.5x$
4	$y = 4.5 \times 27 = 121.5$
5	The hiker takes $121.5$ hours.

Worked example

Error analysis: checking Dev's work

A hiker walks at a steady pace, so her distance is proportional to her time. She walks $9$ kilometers in $2$ hours. How long does she take to walk $27$ kilometers? Dev's five lines of work are in the table above. Find the first wrong line, explain why it is wrong, and correct the answer.

Test lines 1 and 2. Line 1 names both variables with units. The story's complete pair is 2 hours with 9 kilometers, so x = 2 and y = 9. Line 2 divides y by x, in that order, and 2 is not 0. So k = 4.5 kilometers per hour is correct.
$k = \frac{y}{x} = \frac{9}{2} = 4.5$
Test line 3. It has the form y = kx with k = 4.5, so it is the right equation: kilometers equal 4.5 kilometers per hour times hours.
$y = 4.5x$
Test line 4 with the units question. The 27 in the story is 27 kilometers, a distance, so by line 1 it is a y-value. Dev put it in for x, the time. Line 4 is the first wrong line. It answers a different question: how far the hiker walks in 27 hours.
$\text{Dev: } x = 27 \qquad \text{Story: } y = 27$
Confirm with the same-constant question. Dev's answer pairs x = 121.5 hours with y = 27 kilometers. Both numbers are positive and 27 is less than 121.5, so y divided by x is less than 1. It cannot equal 4.5, yet in a proportional relationship every pair where x is not 0 must give k.
$\frac{27}{121.5} \lt 1 \lt 4.5$
Correct line 4. Substitute 27 for y, then divide both sides by 4.5. That keeps the two sides equal because 4.5 is not 0. Multiplying the top and bottom by 10 clears the decimal without changing the value, and 45 times 6 is 270.
$27 = 4.5x \quad \text{so} \quad x = \frac{27}{4.5} = \frac{270}{45} = 6$
Check the corrected pair x = 6, y = 27. It gives the same k as the story's pair, and 6 hours is a positive time, so the answer makes sense.
$\frac{27}{6} = 4.5$

Answer

Line 4 is the first wrong line: Dev substituted $27$ kilometers for $x$ , which stands for time. The correct equation is $27 = 4.5x$ , so the hiker takes $6$ hours to walk $27$ kilometers.

Here is the idea to keep. A proportional word problem like the ones here gives you one complete pair and half of another. The complete pair, with $x \neq 0$ , gives $k = \frac{y}{x}$ . Then $y = kx$ links every pair, so you can put the value you know in place of its own letter and solve for the other one.

The same facts let you judge any solution, yours or someone else's. Match each number to the quantity it measures, make sure $k$ is $y$ divided by $x$ , and test the final pair: when the relationship is proportional and $x \neq 0$ , it must give the same $k$ as the pair in the story.

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Testing and Solving Proportions

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