Finding and Fixing Fraction-Decimal Conversion Errors | Grade 7 Math | LevelKite

A fraction and a decimal can be two ways to write the same number. Converting means changing the form without changing the value. So a conversion is correct only when the fraction and the decimal are the same number.

Suppose a classmate writes $\frac{2}{5} = 2.5$ . Two fifths is less than one whole, but $2.5$ is more than two wholes. The value changed, so something went wrong.

Noticing that is the start of error analysis: you read finished work, find the step where the value changed, explain why that step is wrong, and fix it. In this lesson you will learn three checks that show when a conversion has gone wrong, and you will practice finding the faulty step.

Start with what the symbols mean. A rational number is a number you can write as a fraction $\frac{p}{q}$ , where $p$ and $q$ are integers (whole numbers and their opposites) and $q \neq 0$ . The fraction bar means division: $\frac{p}{q} = p \div q$ , the number that gives $p$ when you multiply it by $q$ . You cannot divide by $0$ , which is why $q \neq 0$ .

So $\frac{2}{5}$ means $2 \div 5$ . It equals $0.4$ , because $0.4 \times 5 = 2$ . Your classmate divided $5$ by $2$ instead.

The decimal form of a rational number comes in only two kinds. To see why, let $p$ and $q$ be positive integers and divide $p$ by $q$ with long division. Each step leaves a remainder, the amount left over.

If a remainder is $0$ , the division stops. You get a terminating decimal, a decimal whose digits stop, such as $\frac{3}{4} = 0.75$ .
If no remainder is ever $0$ , the division never stops. Every remainder is a whole number from $1$ to $q - 1$ , so there are only $q - 1$ possible remainders. (When you divide by $6$ , for example, a remainder that is not $0$ can only be $1$ , $2$ , $3$ , $4$ or $5$ .) One of them must come back after the decimal point, and from then on the same steps and the same digits repeat. You get a repeating decimal, such as $\frac{1}{3} = 0.333\ldots$

You write a repeating decimal exactly by putting a bar over the digits that repeat: $\frac{1}{3} = 0.\overline{3}$ . Stopping after a few digits gives a different number.

One more fact handles negative numbers. Let $p$ and $q$ be integers with $q \neq 0$ . A negative sign can sit in front of the fraction $\frac{p}{q}$ , on its numerator $p$ (the top number), or on its denominator $q$ (the bottom number). All three are the same number.

For example, $-\frac{3}{4}$ , $\frac{-3}{4}$ and $\frac{3}{-4}$ all equal $-0.75$ . Here is why: $(-3) \div 4$ and $3 \div (-4)$ each divide two numbers with different signs, so each answer is negative, and each is the same distance from $0$ as $3 \div 4 = 0.75$ .

So to convert a negative fraction, convert the positive fraction first, then write the negative sign on the decimal. The digits stay the same, so the decimal still terminates or repeats. In symbols, the sign rule is:

-\frac{p}{q}=\frac{-p}{q}=\frac{p}{-q}\qquad\text{for integers }p\text{ and }q\text{ with }q\neq 0

Key concept

A correct conversion keeps the same number

A fraction and a decimal match only when they are the same number. Let $p$ and $q$ be positive integers. Test any claimed conversion with three checks:

Sign: $\frac{p}{q}$ is positive. $-\frac{p}{q}$ , $\frac{-p}{q}$ and $\frac{p}{-q}$ are negative, so their decimal needs a negative sign.
Size: $\frac{p}{q} < 1$ when $p < q$ , and $\frac{p}{q} > 1$ when $p > q$ .
Exact value: a terminating decimal $d$ equals $\frac{p}{q}$ when $d \times q = p$ , and only then. If the division never reaches remainder $0$ , the digits repeat and need a bar. A cut-off copy fails the $d \times q = p$ test.

Failing a check proves there is an error. Passing the sign and size checks does not prove the conversion is right, so always finish with the exact-value check.

d=\frac{p}{q}\quad\text{exactly when}\quad d\times q=p\qquad\text{for integers }p\text{ and }q\text{ with }q\neq 0

Four claims, each tested with one check

Claim to test	Check	What the check shows	Correct conversion
$\frac{-7}{10} = 0.7$	Sign	$\frac{-7}{10} = -\frac{7}{10}$ is negative, but $0.7$ is positive.	$\frac{-7}{10} = -0.7$ , since $0.7 \times 10 = 7$
$\frac{5}{4} = 0.8$	Size	$5 > 4$ , so $\frac{5}{4} > 1$ , but $0.8 < 1$ . The division was done as $4 \div 5$ .	$\frac{5}{4} = 1.25$ , since $1.25 \times 4 = 5$
$\frac{1}{5} = 0.5$	Exact value	$0.5 \times 5 = 2.5$ , not $1$ .	$\frac{1}{5} = 0.2$ , since $0.2 \times 5 = 1$
$\frac{9}{20} = 0.45$	Exact value	$0.45 \times 20 = 9$ , so the two forms are the same number.	No error: $\frac{9}{20} = 0.45$

Worked example

A place-value slip

Maya converts $0.06$ to a fraction. She writes $0.06 = \frac{6}{10} = \frac{3}{5}$ . Find her error, explain it, and fix it.

Run the quick checks first. Both numbers are positive, so the sign check passes. The decimal 0.06 is less than 1, and 3 fifths is less than 1 because 3 is less than 5, so the size check passes too. Passing these two checks does not prove the conversion is right.
$0 < 0.06 < 1 \qquad 0 < \frac{3}{5} < 1$
Run the exact-value check. If 0.06 equals 3 fifths, then 0.06 times 5 must equal 3. It equals 0.3, so the two forms are different numbers. There is an error.
$0.06 \times 5 = 0.3 \neq 3$
Find the step where the value changed. The last digit of 0.06 is in the hundredths place, so 0.06 means 6 hundredths. Maya wrote 6 tenths, which is 0.6. Her first step is the error. Her second step, simplifying 6 tenths to 3 fifths, is correct.
$0.06 = \frac{6}{100} \qquad \frac{6}{10} = 0.6$
Fix it. Write 6 hundredths as a fraction, then divide the numerator and the denominator by 2. Dividing both by the same nonzero number does not change the value of a fraction.
$0.06 = \frac{6}{100} = \frac{6 \div 2}{100 \div 2} = \frac{3}{50}$
Check the new answer with the exact-value check: 0.06 times 50 must equal 3, and it does.
$0.06 \times 50 = 3$

Answer

Maya used tenths, but the digit $6$ is in the hundredths place. The correct conversion is $0.06 = \frac{6}{100} = \frac{3}{50}$ .

Worked example

A skipped zero in long division

Leo converts $\frac{3}{40}$ to a decimal with long division. His notes say: $40$ does not go into $3$ , and it does not go into $30$ . Then $300 \div 40 = 7$ remainder $20$ , and $200 \div 40 = 5$ remainder $0$ . He writes $\frac{3}{40} = 0.75$ . Find his error, explain it, and fix it.

Run the quick checks. Both numbers are positive and both are less than 1, so the sign and size checks pass. Now run the exact-value check: 0.75 times 40 must equal 3. It equals 30, so Leo's decimal is 10 times too large.
$0.75 \times 40 = 30 \neq 3$
Find the step where the value changed. Each long-division step fills one decimal place, even when the digit is 0. Leo's 3 ones are 30 tenths, and 30 divided by 40 gives 0 tenths, so the tenths place needs a 0. Leo wrote nothing there, so his 7 and his 5 each landed one place too far to the left.
$30 \text{ tenths} \div 40 = 0 \text{ tenths, remainder } 30 \text{ tenths}$
Fix it by filling every place in order. The ones place gets 0 and the tenths place gets 0, because 3 and 30 are each less than 40. The 30 tenths left over are 300 hundredths, and 300 divided by 40 gives 7 with 20 left over, so the hundredths place gets 7.
$300 = 40 \times 7 + 20$
The 20 hundredths left over are 200 thousandths, and 200 divided by 40 gives 5 with remainder 0, so the thousandths place gets 5. A remainder of 0 stops the division, so the decimal terminates.
$200 = 40 \times 5 + 0 \qquad \frac{3}{40} = 0.075$
Check the new answer with the exact-value check: 0.075 times 40 must equal 3, and it does.
$0.075 \times 40 = 3$

Answer

Leo skipped the $0$ in the tenths place. The correct conversion is $\frac{3}{40} = 0.075$ .

Close is not equal. A common mistake is to stop a repeating decimal after a few digits, or to round it, and still write an equals sign: $\frac{2}{3} = 0.66$ or $\frac{2}{3} = 0.67$ . Test both with the exact-value check: $0.66 \times 3 = 1.98$ , which is less than $2$ , and $0.67 \times 3 = 2.01$ , which is more than $2$ . So $\frac{2}{3}$ lies between $0.66$ and $0.67$ and equals neither. A rounded decimal can serve as an estimate, but it is not equal to the fraction. More digits do not fix this. When you divide $2$ by $3$ , every step after the decimal point is $20 \div 3 = 6$ remainder $2$ , so the remainder is never $0$ and the digit $6$ repeats forever. The exact decimal is $0.\overline{6}$ . The bar marks the repeat instead of cutting it off.

Worked example

Two errors in one line

Priya converts $-\frac{5}{6}$ to a decimal. She writes $-\frac{5}{6} = 0.83$ . Find every error, explain each one, and fix the conversion.

Run the sign check. Negative 5 sixths can also be written with the negative sign on the 5 or on the 6. It is one number, and it is negative because 5 and 6 are positive. Priya's decimal is positive. A negative number cannot equal a positive number, so the missing negative sign is the first error.
$-\frac{5}{6} = \frac{-5}{6} = \frac{5}{-6} < 0 < 0.83$
Set the sign aside and test the digits: does 0.83 equal 5 sixths? If it does, 0.83 times 6 must equal 5. It equals 4.98, so 0.83 is close to 5 sixths but not equal to it. That is the second error.
$0.83 \times 6 = 4.98 \neq 5$
Divide 5 by 6 to find the exact digits. The ones place gets 0 because 5 is less than 6. Then 5 ones are 50 tenths, and 50 divided by 6 gives 8 with 2 left over. Those 2 tenths are 20 hundredths, and 20 divided by 6 gives 3 with 2 left over.
$50 = 6 \times 8 + 2 \qquad 20 = 6 \times 3 + 2$
The remainder 2 has come back, so the next step is 20 divided by 6 again, and so is every step after it. The remainder is never 0, so the digit 3 repeats forever. Priya stopped after two digits, which cut the repeat off.
$5 \div 6 = 0.8333\ldots$
Mark the repeat with a bar. Only the 3 repeats, and the 8 appears once, so the bar goes over the 3 alone. Then write the negative sign, because negative 5 sixths is the opposite of 5 sixths.
$-\frac{5}{6} = -0.8\overline{3}$

Answer

Priya dropped the negative sign and cut off a repeating decimal. The correct conversion is $-\frac{5}{6} = -0.8\overline{3}$ .

Here is the idea to keep. Converting changes how a rational number is written, never the number itself. So when you check any conversion between a fraction and a decimal, ask three questions:

Is the sign right?
Is the size sensible?
Is the value exact? A terminating decimal times the denominator must give the numerator, and a repeat must be marked with a bar, not cut off.

If any answer is no, find the step where the value changed, explain why it is wrong, and fix it.