Writing Decimals as Fractions
Your video game says you have explored $0.\overline{6}$ of the map. That decimal never ends, so what fraction of the map is it, exactly? Every terminating or repeating decimal can be written as a fraction of whole numbers. On this page you learn to find that fraction.
Kinds of Decimals
Before you convert a decimal, look at how it ends. There are three kinds.
The bar goes over the digits that repeat. You may also see a little hat over those digits instead of a bar.
Every decimal has up to three parts: the whole part (left of the point), the non-repeating part (digits after the point that never repeat) and the repeating part (the block that repeats).
To start working with generating fractions, it is essential to know what type of number we are dealing with. Classify the following decimal numbers into terminating…
To start working with generating fractions, it is essential to know what type of number we are dealing with. Classify the following decimal numbers into terminating decimal, pure repeating decimal, or mixed repeating decimal:
- a) $3,15$
- b) $0,666...$
- c) $12,4777...$
- d) $5,232323...$
- e) $8,02333...$
Identify the parts of the following repeating decimals. For each one, indicate its whole number part, its non-repeating decimal part (if it has one), and its repeating…
Identify the parts of the following repeating decimals. For each one, indicate its whole number part, its non-repeating decimal part (if it has one), and its repeating part:
- a) $5,888...$
- b) $0,1232323...$
- c) $14,5262626...$
Terminating Decimals
A terminating decimal is already a fraction in disguise. Read it out loud: $0.75$ is "seventy-five hundredths," which is $\frac{75}{100}$.
A loading bar is a nice picture of this. If a mission is $0.85$ complete, the bar has $85$ out of $100$ parts filled, and $\frac{85}{100} = \frac{17}{20}$. In the bar below, each block is $\frac{1}{20}$ of the whole, and $17$ of the $20$ blocks are filled.
Calculate the generating fraction of the following terminating decimal numbers. Remember to simplify the result whenever possible: a) $0,5$ b) $1,2$ c) $0,25$ d) $0,08$
Calculate the generating fraction of the following terminating decimal numbers. Remember to simplify the result whenever possible:
- a) $0,5$
- b) $1,2$
- c) $0,25$
- d) $0,08$
In a music store offer, a digital album costs $\$9,90…
In a music store offer, a digital album costs $\$9,90$. Express the price of the album as a fraction of natural numbers and simplify it until it cannot be reduced any further.
In a strategy video game, a loading bar indicates that $0,85$ of a special mission has been completed. What fraction in simplest form of the mission has already been…
In a strategy video game, a loading bar indicates that $0,85$ of a special mission has been completed. What fraction in simplest form of the mission has already been completed? Represent the situation using the following graph and calculate the generating fraction.
Pure Repeating Decimals
Now the endless ones. We can't write "$0.666\ldots$ over $1$ followed by zeros," so we use a trick: subtract the decimal from a bigger copy of itself so the endless tails cancel.
Notice the pattern: the numerator is the repeating digit and the denominator is a $9$. That is always how it goes.
Here is $\frac{2}{3}$ as a bar. The shaded part is the same amount as $0.\overline{6}$ of the whole.
A video game indicates that you have completed $0,\overline{6}$ of a mission. To find out the exact part of the map you have explored, convert this pure repeating…
A video game indicates that you have completed $0,\overline{6}$ of a mission. To find out the exact part of the map you have explored, convert this pure repeating decimal number into its generating fraction.
Hint: Use the rule of placing as many nines in the denominator as there are digits in the repeating part.
Find the generating fraction of the following pure repeating decimal numbers with a single digit in the repeating part. Simplify if you can: a) $0,\overline{3}$ b)…
Find the generating fraction of the following pure repeating decimal numbers with a single digit in the repeating part. Simplify if you can:
- a) $0,\overline{3}$
- b) $0,\overline{7}$
- c) $1,\overline{1}$
- d) $0,\overline{5}$
Find the generating fraction of the pure repeating decimal $12,\widehat{3}$. To do this, follow the algebraic method by calling the number $x$ and multiplying by the…
Find the generating fraction of the pure repeating decimal $12,\widehat{3}$. To do this, follow the algebraic method by calling the number $x$ and multiplying by the appropriate power of 10 to subtract the expressions and eliminate the decimal part. Do not forget to simplify the result.
Mixed Repeating Decimals
When some digits come before the repeat, you need one more step. Let's see why with $x = 0.1\overline{6} = 0.1666\ldots$
Multiply by $10$ to move past the non-repeating digit: $10x = 1.666\ldots$ Multiply by $100$ to move past one full repeat: $100x = 16.666\ldots$ The tails match, so subtract: $100x - 10x = 90x = 16 - 1 = 15$. Then $x = \frac{15}{90} = \frac{1}{6}$.
The whole part tells you which two integers the number sits between. For $5.2\overline{3} = \frac{157}{30}$, that is between $5$ and $6$.
Classify the following decimal numbers into terminating, pure repeating, or mixed repeating. Once classified, calculate the generating fraction of each of them and…
Classify the following decimal numbers into terminating, pure repeating, or mixed repeating. Once classified, calculate the generating fraction of each of them and simplify the result to obtain the fraction in simplest form:
- a) $3,15$
- b) $0,\widehat{12}$
- c) $2,1\widehat{6}$
A young athlete recorded a time of $9,4\widehat{6}$ seconds in a 100-meter sprint. Identify the whole number part, the non-repeating decimal part, and the repeating part…
A young athlete recorded a time of $9,4\widehat{6}$ seconds in a 100-meter sprint. Identify the whole number part, the non-repeating decimal part, and the repeating part of this number. Then, calculate its generating fraction in simplest form to express the time as a fraction.
Look at the following number line where three points have been marked: $A$, $B$, and $C$. Point $B$ represents the decimal number $1,2\widehat{7}$. Find its generating…
Look at the following number line where three points have been marked: $A$, $B$, and $C$. Point $B$ represents the decimal number $1,2\widehat{7}$. Find its generating fraction in simplest form and explain between which two integers this point lies.
Using the Fractions
Why bother? Fractions are exact. Rounded decimals aren't. Once every number is a fraction, you can add, subtract, multiply, divide and compare them with the fraction rules you already know.
Fractions also settle close comparisons. Which is bigger, $0.\overline{4}$ or $0.45$? Write both as fractions: $\frac{4}{9}$ and $\frac{9}{20}$. The LCM of $9$ and $20$ is $180$, so $\frac{4}{9} = \frac{80}{180}$ and $\frac{9}{20} = \frac{81}{180}$. So $0.45$ is a tiny bit bigger.
To make a baking recipe, we need to add $\frac{1}{3}$ of a pound of flour to $0,1\widehat{6}$ of a pound of sugar. To obtain the exact total weight as a fraction, first…
To make a baking recipe, we need to add $\frac{1}{3}$ of a pound of flour to $0,1\widehat{6}$ of a pound of sugar. To obtain the exact total weight as a fraction, first convert the decimal number into its generating fraction and add the two fractions. Simplify the final result.
An organic farmer has designed an irrigation tank whose base is rectangular. The dimensions of the base are $2,\widehat{6}$ feet long and $1,125$ feet wide. If he wants…
An organic farmer has designed an irrigation tank whose base is rectangular. The dimensions of the base are $2,\widehat{6}$ feet long and $1,125$ feet wide. If he wants to install a protective canvas that covers the entire surface of the base, what must the area of this canvas be? Express the result as a fraction in simplest form of a square foot.
Order the following real numbers from least to greatest. To be able to compare them accurately, transform all the decimals into their generating fractions and find a…
Order the following real numbers from least to greatest. To be able to compare them accurately, transform all the decimals into their generating fractions and find a common denominator:
$$\frac{13}{20} \quad ; \quad 0,6\widehat{5} \quad ; \quad 0,\widehat{65} \quad ; \quad \frac{2}{3} \quad ; \quad 0,65$$Equations and Integer Puzzles
Two kinds of harder problems come up again and again. Both start the same way: convert the decimals to fractions in simplest form.
Equations. To solve $0.\overline{4} \cdot x = 2$, write $\frac{4}{9} \cdot x = 2$. Dividing by a fraction means multiplying by its reciprocal: $x = 2 \div \frac{4}{9} = 2 \cdot \frac{9}{4} = \frac{18}{4} = \frac{9}{2}$.
"Smallest number that makes an integer." A fraction in simplest form times a whole number $n$ is a whole number exactly when $n$ is a multiple of the denominator. So the smallest natural number that works is the denominator.
Find the value of the unknown $x$ in the following equation, first expressing the decimal numbers as fractions: $$0,1\widehat{2} \cdot x = 1,1$$ Hint: Convert both…
Find the value of the unknown $x$ in the following equation, first expressing the decimal numbers as fractions:
$$0,1\widehat{2} \cdot x = 1,1$$Hint: Convert both numbers to their generating fraction and solve for $x$ as if it were a fraction equation.
A television game show proposes the following challenge: "Find the smallest natural number which, when multiplied by the mixed repeating decimal $0,1\widehat{36}$,…
A television game show proposes the following challenge: "Find the smallest natural number which, when multiplied by the mixed repeating decimal $0,1\widehat{36}$, results in an integer." What number are they looking for? Explain the steps followed.
A tanker truck transports $12.000$ gallons of milk. At the first stop, it unloads $0,2\widehat{3}$ of the total. At the second stop, it unloads $\frac{3}{11}$ of what…
A tanker truck transports $12.000$ gallons of milk. At the first stop, it unloads $0,2\widehat{3}$ of the total. At the second stop, it unloads $\frac{3}{11}$ of what was left.
- a) How many gallons are left in the truck after the second stop?
- b) Express the total amount unloaded between the two stops as a repeating decimal indicating whether it is pure or mixed.
Back to the start: $0.\overline{6}$ is a pure repeating decimal with one repeating digit, so it equals $\frac{6}{9} = \frac{2}{3}$. You have explored exactly two thirds of the map.