Finding and Fixing Errors in Multi-Step Problems | Grade 7 Math | LevelKite

You already know how to solve multi-step problems with rational numbers: numbers you can write as a fraction of two integers (whole numbers and their opposites) with a denominator that is not zero, such as $7$ , $\frac{3}{4}$ , $-3.5$ and $0.05$ . In this lesson you switch roles. You check a solution that someone else has finished.

Error analysis means reading a solution line by line, finding the first line that is wrong, explaining why it is wrong, and correcting the work.

Why look for the first wrong line? A multi-step solution is a chain. Each line uses numbers from the problem or results from earlier lines. So a wrong result is carried into every later line that uses it. The first wrong line is where the trouble starts.

How do you decide whether a line is right? Ask two questions about it.

Does the line match the problem? It must use the right quantities (the amounts the problem names), the right operation and the right whole. The whole is the amount that a fraction or percent is taken of. For example, a sales tax is a percent of a price, so the price is the whole, and the tax is added to that price.
Is the line true? What stands on the left of an equals sign must equal what stands on the right. Check the signs, any change of form (such as a percent rewritten as a decimal), the order of operations and the arithmetic.

If either answer is no, the line contains an error. A line can also give a wrong value without making a new mistake, just because it uses a wrong result from an earlier line. Such a line needs a new calculation, but it is not where the error began.

Key concept

Find the first wrong line, explain it, then fix forward

To analyze a finished solution, follow five moves.

Estimate. Use easy numbers to predict roughly how big the answer should be and whether it should be positive or negative.
Trace. Go through the lines in order. For each line ask: does it match the problem, and is it true?
Name the error. Point to the first line that fails and say why it fails.
Fix forward. Correct that line, then recompute every later line that uses its result. Keep asking the two questions, because a solution can contain more than one error.
Check. Compare the new answer with your estimate and with the question that was asked.

Every check uses only what is written: the quantities in the problem, what they mean, and the operations in each line.

Worked example

A total that is far too large

A backpack costs $40$ dollars. Sales tax is $5\%$ of the price. Maya finds the total cost in three lines. Line 1: $5\% = 0.5$ . Line 2 (tax): $40 \times 0.5 = 20$ . Line 3 (total): $40 + 20 = 60$ . Maya says the total cost is $60$ dollars. Find the first error, explain it, and correct the work.

Estimate first. The tax is 5% of the price. A smaller percent of the same price is a smaller amount, and 10% of 40 dollars is 4 dollars. So the tax must be less than 4 dollars, and the total must be between 40 and 44 dollars. Maya's 60 dollars is far outside that range, so look for an error.
$\frac{10}{100} \times 40 = \frac{400}{100} = 4$
Trace Line 1. It rewrites the percent from the problem as a decimal, so ask whether it is true. Percent means out of 100, so 5% is 5 hundredths, which is 0.05. The decimal 0.5 is 50 hundredths, which is 50%. Line 1 is false, so it is the first error: Maya's decimal is ten times too large.
$5\% = \frac{5}{100} = 0.05 \quad \text{but} \quad 0.5 = \frac{50}{100} = 50\%$
Fix forward. Line 2 uses the right operation: the tax is 5% of the price, and you find a percent of an amount by multiplying the amount by the percent written as a decimal. But Line 2 uses the wrong decimal from Line 1, so recompute it with 0.05. The tax is 2 dollars.
$40 \times 0.05 = \frac{40 \times 5}{100} = \frac{200}{100} = 2$
Line 3 also uses the right operation: the total cost is the price plus the tax, so you add. It used the old tax of 20 dollars, so recompute it with the corrected tax of 2 dollars.
$40 + 2 = 42$
Check the new answer. 42 dollars is between 40 and 44 dollars, as the estimate predicted, and it is the total cost that the problem asks for.

Answer

The first error is in Line 1, because $5\%$ equals $0.05$ , not $0.5$ . After the correction, the tax is $2$ dollars and the total cost is $42$ dollars.

Worked example

A temperature that moved the wrong way

At 6 a.m. the temperature is $-2.75$ °C (degrees Celsius). It rises $4.5$ degrees by noon and then falls $6.25$ degrees by midnight. Jon finds the midnight temperature in two lines. Line 1 (noon): $-2.75 + 4.5 = 7.25$ . Line 2 (midnight): $7.25 - 6.25 = 1$ . Jon says the midnight temperature is $1$ °C. Find the first error, explain it, and correct the work.

Estimate first. The fall of 6.25 degrees is larger than the rise of 4.5 degrees, so the temperature must end lower than it started. It must be below −2.75 °C, so it must be negative. Jon's answer of 1 °C is positive, so look for an error.
$6.25 > 4.5$
Trace Line 1. It matches the problem, because a rise is added to the starting temperature. But it is not true. The first 2.75 degrees of the rise only bring the temperature from −2.75 °C up to 0 °C. The remaining degrees, 4.5 minus 2.75, take it up to 1.75 °C. Line 1 is the first error: Jon ignored the negative sign and added 2.75 and 4.5.
$-2.75 + 4.5 = 4.5 - 2.75 = 1.75$
Fix forward. Line 2 matches the problem, because a fall is subtracted. But it starts from the wrong noon temperature, so recompute it from 1.75 °C. The first 1.75 degrees of the fall bring the temperature down to 0 °C. The remaining degrees, 6.25 minus 1.75, take it to 4.5 degrees below zero.
$1.75 - 6.25 = -(6.25 - 1.75) = -4.5$
Check the new answer. −4.5 °C is negative and below the starting temperature of −2.75 °C, as the estimate predicted, and it is the midnight temperature that the problem asks for.
$-4.5 < -2.75$

Answer

The first error is in Line 1, because $-2.75 + 4.5$ equals $1.75$ , not $7.25$ . After the correction, the midnight temperature is $-4.5$ °C.

Five false lines to watch for. Each one fails the question: is it true?

Kind of error	False line	Why it is false	Corrected line
Order of operations	$20 - 4 \times 2.5 = 16 \times 2.5 = 40$	With no parentheses around $20 - 4$ , the multiplication $4 \times 2.5 = 10$ is done before the subtraction.	$20 - 4 \times 2.5 = 20 - 10 = 10$
Subtracting a negative	$-3.5 - (-2) = -5.5$	Subtracting a number is the same as adding its opposite, and the opposite of $-2$ is $2$ .	$-3.5 - (-2) = -3.5 + 2 = -1.5$
Multiplying a mixed number	$3 \times 1\frac{3}{4} = 3\frac{3}{4}$	The $3$ multiplies all of $1\frac{3}{4}$ , which is $\frac{7}{4}$ , not only the $1$ .	$3 \times \frac{7}{4} = \frac{21}{4} = 5\frac{1}{4}$
Adding fractions	$\frac{1}{2} + \frac{1}{3} = \frac{2}{5}$	Halves and thirds are parts of different sizes, so rewrite both as sixths before you add.	$\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$
Multiplying a negative into parentheses	$-2(3 - 1.5) = -6 - 3 = -9$	The $-2$ multiplies $3$ and also $-1.5$ . A negative times a negative is positive: $-2 \times (-1.5) = 3$ .	$-2(3 - 1.5) = -6 + 3 = -3$

Worked example

True arithmetic, wrong whole

Ava has $60$ dollars. She spends $\frac{1}{3}$ of her money on a book. Then she spends $\frac{1}{4}$ of the money she has left on lunch. Ava finds how much money remains in three lines. Line 1 (book): $\frac{1}{3} \times 60 = 20$ . Line 2 (lunch): $\frac{1}{4} \times 60 = 15$ . Line 3 (remaining): $60 - 20 - 15 = 25$ . Ava says $25$ dollars remain. Find the first error, explain it, and correct the work.

Estimate first. Each time, Ava spends only part of the money she has, so the amount that remains must be positive and less than 60 dollars. Her answer of 25 dollars fits, so the estimate raises no alarm. That does not prove the work is right, so trace every line.
Trace Line 1. The book costs one third of her money, and her money is 60 dollars, so the whole is 60 dollars. Taking one third of an amount means multiplying the amount by one third. The line matches the problem, and it is true. Line 1 is correct.
$\frac{1}{3} \times 60 = \frac{60}{3} = 20$
Trace Line 2. It is true, because 60 divided by 4 is 15. But it does not match the problem. Lunch costs one fourth of the money Ava has left after the book, so the whole is not her starting 60 dollars. The whole is the amount left after the book, which is 40 dollars. Line 2 is the first error: it takes the fraction of the wrong whole.
$60 - 20 = 40$
Fix forward. Recompute the lunch cost as one fourth of the 40 dollars that Ava has left after the book. Lunch costs 10 dollars.
$\frac{1}{4} \times 40 = \frac{40}{4} = 10$
Line 3 uses the right operation: spending lowers the amount of money, so each cost is subtracted from the starting 60 dollars. It used the old lunch cost of 15 dollars, so recompute it with the corrected cost of 10 dollars.
$60 - 20 - 10 = 40 - 10 = 30$
Check the new answer. 30 dollars is positive and less than 60 dollars. A second route agrees: Ava spends one fourth of the 40 dollars on lunch, so she keeps the other three fourths of it.
$\frac{3}{4} \times 40 = \frac{120}{4} = 30$

Answer

The first error is in Line 2, because the lunch fraction must be taken of the $40$ dollars left after the book, not of the starting $60$ dollars. After the correction, lunch costs $10$ dollars and $30$ dollars remain.

Takeaway. Error analysis is careful reading, not guessing. Estimate the answer, then trace the lines in order and ask of each one: does it match the problem, and is it true? The first line that fails is where the error began. Say why it fails, correct it, and recompute every later line that uses its result. Then check the new answer against your estimate and against the question that was asked. An estimate can raise an alarm. The line-by-line check is what shows you where the error is and why.