The concept of a decimal number

In a sports shop, the prices of three items have been written in words. Write in your notebook the decimal number that corresponds to each one using digits:

1. a) Swimming goggles costing eight euros and fifteen cents.
2. b) A tennis ball costing two euros and eight cents.
3. c) Running shorts costing twelve euros and fifty cents.

State the place value (units, tenths, hundredths or thousandths) of the digit $7$ in each of the following numbers:

1. a) $15.72$
2. b) $0.007$
3. c) $7.431$
4. d) $2.075$

Expand the following decimal numbers according to their place value units, as in the example:

Example: $4.25 = 4$ Units + $2$ tenths + $5$ hundredths.

1. a) $8.39$
2. b) $12.045$
3. c) $0.614$

In a science laboratory, a group of students measured the mass of a mineral sample using a precision balance. The result obtained is $42.308$ grams.

State what place value each digit represents in this number:

• The digit $4$ represents the ............................

• The digit $2$ represents the ............................

• The digit $3$ represents the ............................

• The digit $0$ represents the ............................

• The digit $8$ represents the ............................

Also write its expanded form using units and decimal fractions.

Look at the following number line and write which decimal number corresponds to each of the letters marked with an orange dot:

Copy into your notebook and use the symbols $<$ or $>$ to compare these pairs of decimal numbers:

1. a) $5.4$ ____ $5.39$
2. b) $0.25$ ____ $0.3$
3. c) $12.07$ ____ $12.071$
4. d) $6.1$ ____ $6.099$

In the school Olympics, the times recorded in the $100$ metres flat final were very close. The judges recorded the following results in seconds:

$12.45$ — $12.408$ — $12.5$ — $12.099$ — $12.41$

Order the runners from fastest (shortest time) to slowest (longest time) using the symbol $<$ and write out in words the decimal number that corresponds to the gold medal winner.

Match each decimal number with its corresponding decimal fraction:

• Numbers: $0.7$ | $0.07$ | $7.7$ | $0.77$

• Fractions: $\frac{77}{100}$ | $\frac{7}{10}$ | $\frac{7}{100}$ | $\frac{77}{10}$

If we divide a square into $100$ equal parts, each part is one hundredth ($0.01$). Look at the following image and write which decimal number represents the orange shaded area:

Aoife is packing her rucksack for a trip. She has bought a bottle of water for $1.25$ €, a bag of nuts for $3.050$ € and a cap for $9.9$ €.

1. a) Write all the prices so that they have the same number of decimal places (adding zeros if necessary).
2. b) Convert the price of the cap into an irreducible fraction.
3. c) How many euro cents does the bag of nuts cost?

Classify the following decimal numbers, indicating whether they are terminating decimals, pure recurring decimals or mixed recurring decimals. For recurring decimals, clearly state what the recurring part is and, if present, the non-recurring part:

1. a) $5.08$
2. b) $2.333...$
3. c) $14.252525...$
4. d) $0.1666...$
5. e) $3.141592...$ (is this last one a recurring decimal number?)

Sometimes fractions help us better understand the type of decimal number we have. Without performing long division, classify the decimal expansion of the following fractions as terminating or recurring:

1. a) $\frac{7}{20}$
2. b) $\frac{4}{9}$
3. c) $\frac{13}{6}$
4. d) $\frac{1}{25}$

To find out whether the fraction $\frac{18}{150}$ results in a terminating decimal, a pure recurring decimal, or an eventually recurring decimal, two students propose different approaches:

• Method A (Seán): Directly carry out the division $18 : 150$ using the traditional paper algorithm until checking if the remainder is zero or if the digits start repeating.

• Method B (Aoife): First simplify the fraction to its irreducible form and then factor the new denominator into prime factors to apply the rule of twos and fives.

1. a) Apply both methods step by step and classify the resulting decimal.
2. b) If instead of that fraction you were asked to classify $\frac{21}{8750}$, which of the two methods would you choose and why? Justify your decision by explaining the relationship between the prime factors of the denominator and the powers of $10$.

We propose studying the decimal value of the fraction $\frac{13}{32}$ by combining estimation and theoretical analysis:

1. a) Estimate: Without performing the division, reason between which two consecutive tenths ($0.1$; $0.2$; $0.3$; $0.4$; $0.5$...) this number must lie. To guide yourself, compare it with simpler benchmark fractions with that same denominator, such as $\frac{8}{32}$ and $\frac{16}{32}$.
2. b) Check: Carry out the division to find the first three decimal places and check whether your estimation was correct.
3. c) Reason: Looking at the prime factorization of the denominator of $\frac{13}{32}$, justify whether the complete decimal expansion will end up having a finite number of digits (terminating) or infinite repeating digits. If it is terminating, deduce how many decimal places it will have in total without needing to continue dividing.

Classify the following fractions according to the type of decimal expansion they generate (Terminating, Pure Recurring or Mixed Recurring) without performing the division, based solely on the prime factorisation of their denominators:

1. a) $\frac{7}{40}$
2. b) $\frac{5}{18}$
3. c) $\frac{11}{21}$
4. d) $\frac{9}{250}$