Exercises: Generating fraction of a decimal number

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:

1. a) $3,15$
2. b) $0,666...$
3. c) $12,4777...$
4. d) $5,232323...$
5. e) $8,02333...$

In the science lab, you weighed a sand sample and the scale showed $0,75$ grams. Write this number as a decimal fraction (with a power of 10 in the denominator) and then simplify it to obtain its generating fraction in simplest form.

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:

1. a) $5,888...$
2. b) $0,1232323...$
3. c) $14,5262626...$

Calculate the generating fraction of the following terminating decimal numbers. Remember to simplify the result whenever possible:

1. a) $0,5$
2. b) $1,2$
3. c) $0,25$
4. d) $0,08$

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.

Match each decimal number with its corresponding fraction. It is not necessary to perform long calculations; use the logic of terminating decimals:

Find the generating fraction of the following pure repeating decimal numbers with a single digit in the repeating part. Simplify if you can:

1. a) $0,\overline{3}$
2. b) $0,\overline{7}$
3. c) $1,\overline{1}$
4. d) $0,\overline{5}$

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.

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:

1. a) $3,15$
2. b) $0,\widehat{12}$
3. c) $2,1\widehat{6}$

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.

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.

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.

Match each decimal number in your notebook with its corresponding generating fraction in simplest form. Be careful, there is one fraction that does not correspond to any of the numbers:

A tailor needs to buy fabric to make some costumes. The design requires exactly $2,0\widehat{4}$ yards of blue fabric. In the shop, pieces of fabric are only sold in fractional measurements. Which of the following pieces should the tailor choose if he wants the exact measurement?

1. a) $\frac{202}{99}$ yards.
2. b) $\frac{37}{18}$ yards.
3. c) $\frac{92}{45}$ yards.

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.

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.

Calculate the exact result of the following combined operation. To do this, first find the generating fractions of each decimal number and simplify the intermediate results:

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.

In a laboratory test, a scientist mixes three substances. The first weighs $\frac{5}{9}$ of a gram, the second weighs $0,\widehat{4}$ grams, and the third weighs $0,\widehat{09}$ grams. Show, using generating fractions, that the total sum of the three substances is exactly 1 gram.

Find the value of the unknown $x$ in the following equation, first expressing the decimal numbers as fractions:

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 $0,1\widehat{36}$, results in an integer." What number are they looking for? Explain the steps followed.

Three brothers have inherited an estate and divide it as follows: the eldest gets $0,4\widehat{6}$ of the estate, the middle one gets $0,3\widehat{18}$, and the youngest gets the rest.

1. a) Calculate the generating fraction of the part corresponding to each of the two older brothers.
2. b) What fraction of the total is left for the youngest brother? Express the result as a fraction in simplest form.

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:

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.

1. a) How many gallons are left in the truck after the second stop?
2. b) Express the total amount unloaded between the two stops as a repeating decimal indicating whether it is pure or mixed.