Solve Multi-Step Problems and Check with Estimates | Grade 7 Math | LevelKite

A multi-step problem needs more than one operation to answer. Each step answers a small question, and its result feeds the next step.

The numbers in these problems are rational numbers: numbers that can be written as a fraction of two integers with a nonzero denominator. They can be positive, negative or zero, and they can appear as fractions, mixed numbers, decimals or percents. For example, $-3\frac{1}{4}$ , $7.80$ and 30% are all rational numbers.

Every number in a problem measures something, so before you compute, say what each number means and name its unit. A sign has meaning too. An elevation of $-4.5$ meters means $4.5$ meters below sea level, and a temperature change of $-3$ degrees means a drop of $3$ degrees.

In this lesson you will solve problems like these one step at a time. Then you will use an estimate to check that each answer is reasonable.

Use the same plan for every multi-step problem with rational numbers.

Name the quantities. Say what each number measures, its unit, and what a negative sign means. For a percent, say which amount it is a percent of.
Choose one form. You already know how to convert between fractions, decimals and percents. Decimals are usually easiest for money. Fractions are usually easiest when a decimal would repeat forever, like $\frac{1}{3}$ , or when factors cancel.
Compute one step at a time. Write each step on its own line. Use an operation fact only when its condition holds.
Estimate and compare. Decide whether your answer is reasonable.

Each step you write depends on a fact about operations, and each fact is true only under its condition. The table below lists these facts, where $p$ and $q$ stand for any rational numbers. Four terms appear in it.

The absolute value of a number $q$ , written $\lvert q \rvert$ , is its distance from $0$ .
The opposite of a number, also called its additive inverse, is the number that is the same distance from $0$ on the other side. The opposite of $3.9$ is $-3.9$ , the opposite of $-9$ is $9$ , and the opposite of $0$ is $0$ .
A divisor is the number you divide by. A divisor is never $0$ , because division by $0$ is undefined, which means it has no answer.
The reciprocal of a nonzero number is the number you multiply it by to get $1$ . Every nonzero rational number has one, not only a unit fraction (a fraction with numerator $1$ ) such as $\frac{1}{4}$ . The reciprocal of $\frac{11}{4}$ is $\frac{4}{11}$ , and the reciprocal of $-\frac{3}{2}$ is $-\frac{2}{3}$ . The number $0$ has no reciprocal.

Operation facts for rational numbers

p

and

q

, each with its condition

Fact	Condition	Example
The sum $p + q$ is $\lvert q \rvert$ units from $p$ , in the direction of $q$ .	For all rational $p$ and $q$ : positive direction if $q$ is positive, negative if $q$ is negative, no move if $q = 0$ .	$-4.5 + (-9.75) = -14.25$
A number and its opposite add to $0$ .	True for every rational number.	$3.9 + (-3.9) = 0$
Subtracting a number is adding its opposite: $p - q = p + (-q)$ .	True for every rational $p$ and $q$ .	$-8.5 - (-9) = -8.5 + 9 = 0.5$
When you multiply or divide, the result is negative exactly when an odd count of the numbers are negative.	Every number is nonzero. An even count of negatives, including none, gives a positive result.	$3 \times (-3.25) = -9.75$ and $(-6) \div (-1.5) = 4$
A product with a factor of $0$ is $0$ .	The other factors can be any rational numbers.	$0 \times (-3.25) = 0$
$0$ divided by a nonzero number is $0$ .	The divisor is not $0$ . Division by $0$ is undefined.	$0 \div (-3) = 0$
Dividing by a number is multiplying by its reciprocal.	The divisor is any nonzero rational number: a fraction, mixed number or decimal, positive or negative.	$\frac{3}{4} \div \left(-1\frac{1}{2}\right) = \frac{3}{4} \times \left(-\frac{2}{3}\right) = -\frac{1}{2}$

An estimate is a rough answer that you find quickly, often in your head, by using nearby numbers that are easier to work with. An estimate is not meant to be exact. Its job is to tell you about how large the answer should be, and usually which sign it should have.

To estimate, replace each number with a nearby easier one. Then do the same operations in the same order. Keep the sign of each number, and never round a divisor to $0$ . You can estimate before or after the exact work.

An answer is reasonable when it is close to your estimate and makes sense in the situation. There is no fixed cutoff for close. Ask yourself: could the rounding I did explain the gap? If it could not, recheck both your exact work and your estimate. Two strong warning signs are an answer that is many times bigger or smaller than the estimate, and an answer whose sign is opposite to an estimate that is well away from $0$ .

Three ways to choose easier numbers

Strategy	Exact problem	Estimate	Exact value
Round to a nearby whole number	$6.2 \times (-4.9)$	$6 \times (-5) = -30$	$-30.38$
Use a benchmark, a familiar value such as $\frac{1}{2}$ or 10%	48% of $61.90$	$\frac{1}{2}$ of $62$ is $31$	$29.712$
Pick compatible numbers, nearby numbers that divide evenly	$-23.94 \div 3\frac{4}{5}$	$-24 \div 4 = -6$	$-6.3$

Key concept

Solve step by step, then check with an estimate

A multi-step problem is a chain of small steps. In each step, say what the numbers mean, write them in one form, and use an operation fact only when its condition holds.

Then estimate: work the problem again with nearby, easier numbers. Your answer is reasonable when it is close to the estimate and fits the situation, including its sign.

If the gap is larger than your rounding can explain, recheck both the exact work and the estimate. A close match supports your answer, but it does not prove that every step is correct.

Worked example

Three forms in one problem

You buy $2\frac{1}{2}$ pounds of almonds that cost $7.80$ dollars per pound. A coupon takes 30% off the cost. How much do you pay? Check that your answer is reasonable.

Name the quantities and choose one form. The weight is in pounds, the price is in dollars per pound, and the coupon removes 30% of the cost. The price is already a decimal and the values are money, so write the weight and the percent as decimals.
$2\frac{1}{2} = 2.5, \quad 30\% = 0.30$
Find the cost before the coupon. Each pound costs 7.80 dollars, so multiply the number of pounds by the price per pound. Split 2.5 into 2 and 0.5 to multiply in parts.
$2.5 \times 7.80 = 2 \times 7.80 + 0.5 \times 7.80 = 15.60 + 3.90 = 19.50$
Find the discount. The coupon takes 30% of the cost, so multiply the cost by 0.30. One tenth of 19.50 is 1.95, and 0.30 is three tenths.
$0.30 \times 19.50 = 3 \times 1.95 = 5.85$
Subtract the discount from the cost, because you pay the cost with the discount taken away.
$19.50 - 5.85 = 13.65$
Estimate with easier numbers. Round 7.80 up to 8 and keep 2.5, so the cost is about 20 dollars. Then 30% of 20 is 6, and taking 6 from 20 leaves 14.
$2.5 \times 8 = 20, \quad 0.30 \times 20 = 6, \quad 20 - 6 = 14$
Compare. The exact answer 13.65 is 0.35 below the estimate 14. Rounding the price up by 0.20 dollars per pound explains a gap that small, so the answer is reasonable.
$14 - 13.65 = 0.35$

Answer

You pay $13.65$ dollars. This is close to the estimate of about $14$ dollars, so the answer is reasonable.

Worked example

Negative numbers that carry meaning

A diver is at an elevation of $-4.5$ meters, which means $4.5$ meters below sea level. She descends for $3$ minutes, and her elevation changes by $-3\frac{1}{4}$ meters in each minute. Then she rises $1.6$ meters. What is her final elevation? Check that your answer is reasonable.

Name the quantities. Elevation is height compared with sea level, in meters, so a negative elevation is below sea level. A negative change means going down, and a positive change means going up. Write the mixed number as a decimal to match the other values.
$-3\frac{1}{4} = -3.25$
Find the total change during the descent. Her elevation changes by the same amount in each of the 3 minutes, so multiply. Both factors are nonzero and exactly one is negative. A count of one is odd, so the product is negative.
$3 \times (-3.25) = -(3 \times 3.25) = -9.75$
Add that change to the starting elevation. Adding -9.75 moves 9.75 units from -4.5 in the negative direction. Both distances are below sea level, so they add.
$-4.5 + (-9.75) = -(4.5 + 9.75) = -14.25$
Add the rise. Adding 1.6 moves 1.6 units from -14.25 in the positive direction, toward sea level. She is still below sea level, because 1.6 is less than 14.25.
$-14.25 + 1.6 = -(14.25 - 1.6) = -12.65$
Estimate with whole numbers: start at about -5, change by about -3 in each of the 3 minutes, then rise about 2.
$-5 + 3 \times (-3) + 2 = -5 + (-9) + 2 = -12$
Compare by finding the gap. Subtracting -12.65 is the same as adding its opposite, 12.65. Both values are negative and less than 1 meter apart, a gap that rounding to whole numbers can explain, so the answer is reasonable.
$-12 - (-12.65) = -12 + 12.65 = 0.65$

Answer

The diver's final elevation is $-12.65$ meters, which is $12.65$ meters below sea level. This is close to the estimate of $-12$ meters, so the answer is reasonable.

Worked example

A rate with a mixed-number divisor

In the first $2\frac{3}{4}$ hours of a cold front, the outdoor temperature changes by $-8\frac{1}{4}$ degrees Celsius at a steady rate. The temperature is $5.9$ degrees Celsius when the cold front arrives. If the same rate continues, what is the temperature $4.8$ hours after the cold front arrives? Check that your answer is reasonable.

Define the rate. Let r be the temperature change in each hour, in degrees Celsius per hour. A steady rate is the total change divided by the time it takes. The divisor here is the time, which is not 0, so the quotient is defined.
$r = -8\frac{1}{4} \div 2\frac{3}{4}$
Write both mixed numbers as fractions, then multiply by the reciprocal of the divisor. This is valid for any nonzero divisor, including a mixed number. Both numbers are nonzero and exactly one is negative, an odd count, so the quotient is negative.
$r = -\frac{33}{4} \div \frac{11}{4} = -\frac{33}{4} \times \frac{4}{11} = -\frac{132}{44} = -3$
Find the change over 4.8 hours. The temperature changes by -3 degrees in each hour, so multiply the number of hours by the rate. Both factors are nonzero and exactly one is negative, so the product is negative.
$4.8 \times (-3) = -(4.8 \times 3) = -14.4$
Add the change to the starting temperature. Adding -14.4 moves 14.4 units from 5.9 in the negative direction. The first 5.9 units reach 0, and the other 8.5 units go below 0.
$5.9 + (-14.4) = -(14.4 - 5.9) = -8.5$
Estimate with compatible numbers and compare: about -9 degrees in about 3 hours, then about 5 hours from a start of about 6 degrees. The rounded divisor, 3, is not 0. The exact answer -8.5 and the estimate -9 are both negative and only 0.5 degrees apart, a gap the rounding can explain, so the answer is reasonable.
$-9 \div 3 = -3, \quad 5 \times (-3) = -15, \quad 6 + (-15) = -9$

Answer

The temperature $4.8$ hours after the cold front arrives is $-8.5$ degrees Celsius. This is close to the estimate of $-9$ degrees Celsius, so the answer is reasonable.

Here is the idea to take with you. To solve a multi-step problem with rational numbers, say what each number means, write the numbers in one form, and compute one step at a time, using each operation fact only when its condition holds. Then estimate with nearby easier numbers and compare. An answer that is close to the estimate and fits the situation is reasonable. A gap that your rounding cannot explain tells you to recheck your work and your estimate.

Next, you will use these ideas for error analysis: finding and fixing mistakes in multi-step problems with rational numbers.