Rational Number Word Problems and Error Analysis | Grade 7 Math | LevelKite

In this lesson you use what you know about multiplying and dividing in two ways. First you solve word problems with signed numbers. Then you find and fix mistakes in finished work.

The numbers are rational numbers: numbers you can write as a fraction $\frac{a}{b}$ , where $a$ and $b$ are integers (whole numbers and their opposites) and $b$ is not $0$ . Fractions, mixed numbers and decimals that end or repeat are rational, and so are their opposites.

In a word problem, a sign usually stands for a direction, such as up or down. Suppose you choose up as positive. Then a diver who goes down $2$ meters has a change in elevation of $-2$ meters, where elevation means height compared with a zero level. So a signed answer has two parts to read: its size, called its absolute value (its distance from $0$ ), and its direction, shown by its sign.

Many word problems describe a steady change, where the same change happens again and again. A rate tells how much one quantity changes for each $1$ unit of another, such as meters per second or degrees per hour. A rate is a quotient (the result of a division): a change divided by a nonzero number of units.

When the rate stays the same, the total change is a product (the result of a multiplication): the rate times the number of units, such as the number of seconds. For example, a change of $-2$ meters each second for $3$ seconds gives $-2 \times 3 = -6$ meters in all.

\text{total change} = \text{rate} \times \text{number of units}

That one relationship answers three kinds of question. Division undoes multiplication, so when you know the total change and one of the other two numbers, you divide the total change by the number you know, as long as that number is not $0$ . The number you divide by can be any rational number except $0$ : a fraction, a mixed number or a decimal, positive or negative. The table shows which operation each question needs.

Which operation does the question need?

You know	You want	Operation
The rate and the number of units	The total change	Multiply: rate $\times$ number of units
The total change and the number of units (not $0$ )	The rate	Divide: total change $\div$ number of units
The total change and the rate (not $0$ )	The number of units	Divide: total change $\div$ rate

Key concept

Sign, size, and the multiply-back check

A signed answer has a size and a direction, and you can work them out separately.

Sign. Each negative number reverses the direction once. So when none of the numbers is $0$ , a product or quotient is negative exactly when an odd number of the numbers you multiply or divide are negative. Otherwise it is positive.

Zero. If any factor (a number being multiplied) is $0$ , the product is $0$ . Zero divided by a nonzero number is $0$ . Division by $0$ is undefined.

Check. In $a \div b = c$ , $a$ is the dividend, $b$ is the divisor and $c$ is the quotient. When $b$ is not $0$ , the quotient times the divisor must give back the dividend. If it does not, the work contains an error.

a \div b = c \quad \text{means} \quad c \times b = a \qquad (b \neq 0)

Worked example

Multiplying to find a total change

A hot-air balloon is coming down at a steady rate. Its elevation changes by $-2.4$ meters each second. What is the total change in its elevation after $3.5$ seconds?

Name the quantities. Take up as the positive direction, so a falling elevation is a negative change. Let r be the change in elevation each second, and let t be the number of seconds.
$r = -2.4 \text{ meters per second}, \quad t = 3.5 \text{ seconds}$
Choose the operation. The same change happens every second, so the total change is the rate times the number of seconds.
$r \times t = -2.4 \times 3.5$
Find the sign. Neither factor is 0, and exactly one factor is negative. One is an odd count, so the product is negative.
Find the size by multiplying the absolute values. Split 3.5 into 3 and 0.5, then use the distributive property to multiply each part by 2.4 and add the results.
$2.4 \times 3.5 = 2.4 \times 3 + 2.4 \times 0.5 = 7.2 + 1.2 = 8.4$
Put the sign and the size together. Then check the sense: meters per second times seconds gives meters, and a balloon that is coming down should have a negative change in elevation.
$-2.4 \times 3.5 = -8.4$

Answer

The balloon's elevation changes by $-8.4$ meters, so the balloon is $8.4$ meters lower after $3.5$ seconds.

Worked example

Dividing by a negative fraction

A water tank is draining at a steady rate. Its water level changes by $-\frac{3}{4}$ inch each hour. How many hours does it take for the level to change by $-4\frac{1}{2}$ inches?

Name the quantities. Take a rising level as positive, so both changes here are negative. Let r be the change in level each hour, let c be the total change, and let h be the number of hours.
$r = -\frac{3}{4} \text{ inch per hour}, \quad c = -4\frac{1}{2} \text{ inches}$
Choose the operation. The rate times the number of hours equals the total change. The rate is not 0, so dividing the total change by the rate gives the number of hours.
$h = c \div r = -4\frac{1}{2} \div \left(-\frac{3}{4}\right)$
Write the mixed number as a fraction so that you can multiply. Four and one half is 8 halves plus 1 half, which is 9 halves. The negative sign stays.
$-4\frac{1}{2} = -\frac{9}{2}$
Dividing by a nonzero number is the same as multiplying by its reciprocal, the number whose product with it is 1. The reciprocal of negative three fourths is negative four thirds.
$-\frac{9}{2} \div \left(-\frac{3}{4}\right) = -\frac{9}{2} \times \left(-\frac{4}{3}\right)$
Find the sign and the size. No factor is 0 and two factors are negative. Two is an even count, so the product is positive. Multiply the numerators, multiply the denominators, then simplify.
$-\frac{9}{2} \times \left(-\frac{4}{3}\right) = \frac{9 \times 4}{2 \times 3} = \frac{36}{6} = 6$
Check by multiplying back. The divisor is not 0, so the quotient times the divisor must return the dividend, which is the total change.
$6 \times \left(-\frac{3}{4}\right) = -\frac{18}{4} = -\frac{9}{2} = -4\frac{1}{2}$

Answer

It takes $6$ hours for the water level to change by $-4\frac{1}{2}$ inches.

A tempting mistake. In the tank problem the level is falling and both numbers are negative, so it can feel as if the answer must be negative too: $-6$ hours. That reasoning fails because the quotient does not measure the water level. It answers a different question: what number times $-\frac{3}{4}$ gives $-4\frac{1}{2}$ ? Test $-6$ : you get $-6 \times \left(-\frac{3}{4}\right) = 4\frac{1}{2}$ , a level that rises, which does not match the story. Test $6$ : you get $6 \times \left(-\frac{3}{4}\right) = -4\frac{1}{2}$ , which does match. For nonzero numbers, the sign of a product or quotient comes from counting the negative numbers in the calculation (odd count: negative; even count: positive), not from how the story sounds.

Error analysis means reading a finished solution one line at a time. You look for the first line that does not follow from the line before it, explain why it fails, and correct the work from that point on. A wrong final answer only tells you that an error exists. The lines tell you where it is.

Ask four questions as you read:

Setup. Do the numbers, signs and units match the story? Does the operation match the question, and is any division by a nonzero number?
Sign. When no number is $0$ , is the result negative exactly when an odd number of the numbers multiplied or divided are negative?
Size. Is the arithmetic correct? In a division, was only the divisor replaced by its reciprocal?
Sense. Does the answer fit the story? For a division with a nonzero divisor, does the quotient times the divisor give back the dividend?

Five common mistakes, why each one fails, and the correct work

Mistaken work	Why it fails	Correct work
$-0.6 \times (-1.5) = -0.9$	Neither factor is $0$ and two are negative. Two is an even count, so the product is positive.	$-0.6 \times (-1.5) = 0.9$
$(-2)(-1.5)\left(-\frac{1}{3}\right) = 1$	No factor is $0$ and three are negative. Three is an odd count, so the product is negative.	$(-2)(-1.5)\left(-\frac{1}{3}\right) = 3 \times \left(-\frac{1}{3}\right) = -1$
$\frac{5}{6} \div \left(-\frac{2}{3}\right) = \frac{6}{5} \times \left(-\frac{2}{3}\right)$	The reciprocal of the dividend was used. Only the nonzero divisor is replaced by its reciprocal.	$\frac{5}{6} \times \left(-\frac{3}{2}\right) = -\frac{15}{12} = -\frac{5}{4}$
$-7.2 \div 0 = 0$	Division by $0$ is undefined: no number times $0$ equals $-7.2$ .	$-7.2 \div 0$ is undefined
$0 \div (-7.2)$ is undefined	Zero divided by a nonzero number is $0$ , because $0 \times (-7.2) = 0$ .	$0 \div (-7.2) = 0$

Worked example

Error analysis: find the first wrong line

A cave explorer climbs down at a steady rate. Her elevation changes by $-5\frac{1}{4}$ meters in $1\frac{3}{4}$ minutes. Leo finds the change per minute like this: $-5\frac{1}{4} \div 1\frac{3}{4} = -\frac{21}{4} \div \frac{7}{4} = -\frac{4}{21} \times \frac{7}{4} = -\frac{28}{84} = -\frac{1}{3}$ . He says her elevation changes by $-\frac{1}{3}$ meter each minute. Find Leo's first error and correct it.

Check the setup. Take up as positive, so the total change is negative. The change per minute is the total change divided by the number of minutes, and that divisor is not 0. Leo's first expression is correct.
$-5\frac{1}{4} \div 1\frac{3}{4}$
Check the mixed numbers. Five and one fourth is 20 fourths plus 1 fourth, or 21 fourths. One and three fourths is 4 fourths plus 3 fourths, or 7 fourths. This step is correct too.
$-5\frac{1}{4} = -\frac{21}{4}, \qquad 1\frac{3}{4} = \frac{7}{4}$
Check the change to multiplication. Dividing by a nonzero number means multiplying by the reciprocal of the divisor, and the dividend stays as it is. Leo took the reciprocal of the dividend and left the divisor unchanged, so this line is his first error.
$-\frac{21}{4} \div \frac{7}{4} \neq -\frac{4}{21} \times \frac{7}{4}$
Correct the work from that line. Keep the dividend and multiply by four sevenths, the reciprocal of the divisor. No factor is 0 and one factor is negative. One is an odd count, so the product is negative.
$-\frac{21}{4} \times \frac{4}{7} = -\frac{84}{28} = -3$
Check by multiplying back. The divisor is not 0, so the change per minute times the number of minutes must return the total change. The corrected value does. Leo's value does not.
$-3 \times \frac{7}{4} = -\frac{21}{4} = -5\frac{1}{4}, \qquad -\frac{1}{3} \times \frac{7}{4} = -\frac{7}{12}$

Answer

Leo's first error is taking the reciprocal of the dividend instead of the divisor. The correct change is $-3$ meters each minute, so the explorer's elevation drops $3$ meters every minute.

Takeaway. A signed answer tells a size and a direction. To solve a word problem, name each quantity with its unit and its positive direction. Then use the fact that total change equals rate times number of units to choose multiplication, or division by a nonzero number. For nonzero numbers, the result is negative exactly when an odd number of the numbers you multiply or divide are negative. To check any solution, yours or someone else's, go line by line through setup, sign, size and sense. When the divisor is not $0$ , the quotient times the divisor must return the dividend. The first line that fails a check shows you where the error is.