Estimate and Solve Fraction Word Problems | Grade 7 Math | LevelKite

You already know how to add, subtract, multiply and divide fractions and mixed numbers. (A mixed number is a whole number plus a fraction, such as $2\frac{3}{4}$ .) This lesson puts those skills to work in two ways.

First, you learn to estimate: to find a number close to the exact answer by using easier numbers. Then you solve word problems, and you use your estimate to check each exact answer.

An estimate takes only a few seconds. It cannot prove that an answer is right. But when your exact answer is far from your estimate, you know to look again at the operation you chose and at your arithmetic.

A benchmark is an easy number that you compare other numbers with. For a fraction between $0$ and $1$ , use the benchmarks $0$ , $\frac{1}{2}$ and $1$ . Compare the numerator (the top number) with the denominator (the bottom number, which is never $0$ ):

If the numerator is much smaller than the denominator, the fraction is close to $0$ . Example: $\frac{1}{10}$ .
If the numerator is about half the denominator, the fraction is close to $\frac{1}{2}$ . Example: $\frac{5}{12}$ , because half of $12$ is $6$ .
If the numerator is almost as large as the denominator, the fraction is close to $1$ . Example: $\frac{7}{8}$ .

For a mixed number, round to the nearest whole number. Every quantity in this lesson is positive, so you only need to look at the fraction part:

If it is less than $\frac{1}{2}$ , keep the whole number. So $5\frac{1}{6}$ rounds to $5$ .
If it is more than $\frac{1}{2}$ , go up to the next whole number. So $1\frac{7}{8}$ rounds to $2$ .
If it is exactly $\frac{1}{2}$ , the number is halfway, and either whole number is a fair choice.

After you round, do the operation with the easy numbers in your head. The result is your estimate. The table shows this for all four operations, and its first two rows use the numbers you have just rounded.

Round each number, then operate. In every row the estimate is close to the exact answer.

Operation	Problem	Rounded numbers	Estimate	Exact answer
Sum (result of adding)	$\frac{7}{8} + \frac{5}{12} + \frac{1}{10}$	$1 + \frac{1}{2} + 0$	$1\frac{1}{2}$	$1\frac{47}{120}$
Difference (result of subtracting)	$5\frac{1}{6} - 1\frac{7}{8}$	$5 - 2$	$3$	$3\frac{7}{24}$
Product (result of multiplying)	$\frac{4}{7} \times 5\frac{8}{9}$	$\frac{1}{2} \times 6$	$3$	$3\frac{23}{63}$
Quotient (result of dividing)	$7\frac{2}{3} \div 1\frac{7}{8}$	$8 \div 2$	$4$	$4\frac{4}{45}$

Worked example

Estimate a quotient

Estimate $6\frac{1}{4} \div \frac{5}{9}$ . Then find the exact quotient and compare the two.

Round the dividend, which is the number being divided. Its fraction part, one fourth, is less than one half, so the dividend is closer to 6 than to 7.
$6\frac{1}{4} \approx 6$
Round the divisor. Half of 9 is 4 and a half, and the numerator 5 is close to that, so the divisor is close to one half. One half is not 0, so you can divide by it.
$\frac{5}{9} \approx \frac{1}{2}$
Divide the rounded numbers. Dividing 6 by one half asks how many halves fit into 6. Each whole holds 2 halves, so 6 wholes hold 6 times 2 halves. The estimate is 12.
$6 \div \frac{1}{2} = 6 \times 2 = 12$
Now find the exact quotient. Write the dividend as one fraction: 6 wholes are 24 fourths, and 1 more fourth makes 25 fourths. The divisor is not 0, so dividing by it gives the same result as multiplying by its reciprocal, the same fraction with its top and bottom swapped.
$6\frac{1}{4} \div \frac{5}{9} = \frac{25}{4} \div \frac{5}{9} = \frac{25}{4} \times \frac{9}{5}$
Multiply the numerators and multiply the denominators. Divide the top and bottom by 5 to simplify, then write 45 fourths as 11 wholes and 1 fourth.
$\frac{25}{4} \times \frac{9}{5} = \frac{225}{20} = \frac{45}{4} = 11\frac{1}{4}$
Compare. The exact quotient is less than 1 away from the estimate, so the estimate is a good one.
$12 - 11\frac{1}{4} = \frac{3}{4}$

Answer

A good estimate for $6\frac{1}{4} \div \frac{5}{9}$ is $12$ . The exact quotient is $11\frac{1}{4}$ , which is close to the estimate.

Now put estimating to work in word problems. Most word problems do not name the operation for you. You decide from what happens in the situation. Before you compute:

Name each quantity and its unit, such as meters or hours.
Ask what happens to the quantities. Are they joined, taken away, repeated, or split into equal groups?
Choose the operation that matches.

The table shows what each operation means in a situation.

Match what happens to the operation

What happens	Operation	Example
Two amounts are joined into one total.	Add	Walk $1\frac{1}{2}$ km, then $2\frac{1}{4}$ km: $1\frac{1}{2} + 2\frac{1}{4} = 3\frac{3}{4}$ km in all.
One amount is taken away from another, or you find how much more one amount is than another.	Subtract	Cut $1\frac{3}{4}$ m from $4\frac{1}{2}$ m of rope: $4\frac{1}{2} - 1\frac{3}{4} = 2\frac{3}{4}$ m left.
You take a fraction of an amount, or you repeat equal groups.	Multiply	Use $\frac{2}{3}$ of $4\frac{1}{2}$ cups of flour: $\frac{2}{3} \times 4\frac{1}{2} = 3$ cups.
You find how many equal groups fit, or how much there is for each 1 unit.	Divide (the divisor is not $0$ )	Pour $4\frac{1}{2}$ cups into $\frac{3}{4}$ -cup servings: $4\frac{1}{2} \div \frac{3}{4} = 6$ servings.

Key concept

Estimate, solve, compare

An estimate is a quick answer you find with easy numbers that are close to the real ones. Use it in every fraction problem:

Estimate. Round each fraction between $0$ and $1$ to $0$ , $\frac{1}{2}$ or $1$ , and round each mixed number to the nearest whole number. Never round a divisor to $0$ , because dividing by $0$ is not defined.
Solve. Let the situation choose the operation, then compute the exact answer.
Compare. If the exact answer is far from the estimate, check the operation and the arithmetic again.

Mistake to avoid: thinking that dividing always makes a number smaller. Suppose $4\frac{3}{8}$ pounds of trail mix go into $\frac{5}{8}$ -pound bags. A student expects fewer than $4\frac{3}{8}$ bags and multiplies instead: $4\frac{3}{8} \times \frac{5}{8} = \frac{35}{8} \times \frac{5}{8} = \frac{175}{64} = 2\frac{47}{64}$ .

This fails because each bag holds less than $1$ pound, so every pound fills more than one bag. A quick estimate, $4 \div \frac{1}{2} = 8$ , exposes the error. The exact count is $\frac{35}{8} \div \frac{5}{8} = \frac{35}{8} \times \frac{8}{5} = 7$ bags.

For a positive amount, dividing by a number between $0$ and $1$ gives a larger result, and dividing by a number greater than $1$ gives a smaller one.

Worked example

Ribbon left on a roll

A roll holds $10\frac{2}{3}$ m of ribbon. You cut off one piece that is $3\frac{3}{4}$ m long and another piece that is $2\frac{5}{6}$ m long. How many meters of ribbon are left on the roll?

Name the quantities and choose the operations. Let u be the meters of ribbon you cut off in all, and let r be the meters left on the roll. The two pieces are joined into one total, so add them. That total is taken away from the roll, so subtract it.
$u = 3\frac{3}{4} + 2\frac{5}{6}, \qquad r = 10\frac{2}{3} - u$
Estimate first. Each fraction part here is more than one half, so each mixed number rounds up to the next whole number. About 7 meters are cut off, and about 4 meters are left.
$u \approx 4 + 3 = 7, \qquad r \approx 11 - 7 = 4$
Add exactly. Use twelfths, because 12 is the smallest number that 4 and 6 both divide evenly. Multiplying the top and bottom of a fraction by the same nonzero number keeps its value, so three fourths is 9 twelfths and five sixths is 10 twelfths. The 19 twelfths regroup as 1 whole and 7 twelfths.
$u = 3\frac{3}{4} + 2\frac{5}{6} = 3\frac{9}{12} + 2\frac{10}{12} = 5\frac{19}{12} = 6\frac{7}{12}$
Subtract exactly. Multiply the top and bottom of two thirds by 4 to write the roll's length in twelfths. Subtract the whole numbers, then the twelfths. No regrouping is needed, because 8 twelfths is more than 7 twelfths.
$r = 10\frac{2}{3} - 6\frac{7}{12} = 10\frac{8}{12} - 6\frac{7}{12} = 4\frac{1}{12}$
Compare with the estimate. The exact answer is only one twelfth of a meter more than the estimate of 4 meters, so it is reasonable.
$4\frac{1}{12} - 4 = \frac{1}{12}$

Answer

There are $4\frac{1}{12}$ meters of ribbon left on the roll.

Worked example

A steady hiking speed

Ana hikes $6\frac{3}{4}$ miles in $1\frac{4}{5}$ hours. She keeps the same steady speed. How far does she hike in $2\frac{2}{3}$ hours?

Name the quantities. Let s be Ana's speed in miles per hour, and let d be the miles she hikes in the second, longer time. Speed is the distance for each 1 hour, so divide the miles by the hours. The time is not 0, so this quotient exists. At a steady speed every hour adds the same s miles, so multiply s by the new time.
$s = 6\frac{3}{4} \div 1\frac{4}{5}, \qquad d = s \times 2\frac{2}{3}$
Estimate first. Each fraction part is more than one half, so the three mixed numbers round up to 7, 2 and 3. Divide, then multiply, with these easy numbers.
$s \approx 7 \div 2 = 3\frac{1}{2}, \qquad d \approx 3\frac{1}{2} \times 3 = 10\frac{1}{2}$
Divide exactly. Write each mixed number as one fraction: 27 fourths and 9 fifths. The divisor is not 0, so dividing by 9 fifths gives the same result as multiplying by its reciprocal, 5 ninths. Divide the top and bottom of the product by 9 to simplify, then write 15 fourths as 3 wholes and 3 fourths.
$s = \frac{27}{4} \div \frac{9}{5} = \frac{27}{4} \times \frac{5}{9} = \frac{135}{36} = \frac{15}{4} = 3\frac{3}{4}$
Multiply exactly. Use the fraction form of the speed, and write the new time as 8 thirds. Multiply the numerators, multiply the denominators, and simplify: 120 divided by 12 is 10.
$d = \frac{15}{4} \times \frac{8}{3} = \frac{120}{12} = 10$
Compare with the estimates and check the unit. The exact speed is one fourth away from its estimate, and the exact distance is one half away, so both are reasonable. Miles per hour times hours gives miles, the unit the question asks for.
$3\frac{3}{4} - 3\frac{1}{2} = \frac{1}{4}, \qquad 10\frac{1}{2} - 10 = \frac{1}{2}$

Answer

Ana's speed is $3\frac{3}{4}$ miles per hour, so she hikes $10$ miles in $2\frac{2}{3}$ hours.

The takeaway: an estimate with easy numbers is a quick check on your exact work. Round fractions between $0$ and $1$ to $0$ , $\frac{1}{2}$ or $1$ , and round mixed numbers to the nearest whole number, but never round a divisor to $0$ . In a word problem, let the situation choose the operation: joining means add, taking away or finding how much more means subtract, a fraction of an amount or equal groups means multiply, and finding how many equal groups fit or the amount for each 1 unit means divide. Then estimate, solve exactly, and compare. A large gap tells you to check your work again.