Decimal fractions (tenths, hundredths) Knowledge-Based Study

Study notes

1. What is a Tenth?

   A decimal fraction is a fraction whose denominator is a power of 10, such as 10 or 100. When a whole unit is divided into 10 equal parts, each part represents one tenth.

   • In fractional form, one tenth is written as $\frac{1}{10}$.
   • In decimal form, one tenth is written as 0.1.

   The place value immediately to the right of the decimal point is the tenths place. For example, in the number 0.7, the digit 7 represents seven tenths, or $\frac{7}{10}$.

   When a number includes whole units and tenths, it can be written as a mixed number or a decimal:

   • $2\frac{4}{10} = 2.4$ (read as "two and four tenths")
   • $5\frac{9}{10} = 5.9$ (read as "five and nine tenths")

   What is a Hundredth?

   When a whole unit is divided into 100 equal parts, each part represents one hundredth.

   • In fractional form, one hundredth is written as $\frac{1}{100}$.
   • In decimal form, one hundredth is written as 0.01.

   The second place value to the right of the decimal point is the hundredths place. For instance, in the number 0.08, the digit 8 represents eight hundredths, or $\frac{8}{100}$.

   When both tenths and hundredths are present:

   • $\frac{35}{100} = 0.35$ (read as "thirty-five hundredths")
   • $0.35$ consists of 3 tenths and 5 hundredths: $\frac{3}{10} + \frac{5}{100} = \frac{30}{100} + \frac{5}{100} = \frac{35}{100}$.
   • $4\frac{12}{100} = 4.12$ (read as "four and twelve hundredths")

   Relating Tenths to Hundredths

   Tenths and hundredths are closely related because 1 tenth is equal to 10 hundredths:

   $$\frac{1}{10} = \frac{10}{100} = 0.1 = 0.10$$

   Multiplying the numerator and denominator of a tenth fraction by 10 gives an equivalent fraction in hundredths:

   • $\frac{6}{10} = \frac{60}{100} = 0.6 = 0.60$
   • $\frac{9}{10} = \frac{90}{100} = 0.9 = 0.90$

   Common Mistake: Do not confuse tenths with single-digit hundredths. For example, $\frac{4}{10} = 0.4$, whereas $\frac{4}{100} = 0.04$. The zero after the decimal point in 0.04 holds the tenths place, ensuring the 4 remains in the hundredths place.

   How to Compare Tenths and Hundredths

   To compare decimal numbers involving tenths and hundredths, align the digits by place value or convert them to equivalent fractions with a common denominator of 100.

   Step-by-Step Method

   1. Compare whole numbers first: A number with a greater whole number is always larger (e.g., $3.1 > 2.85$).
   2. Pad with a trailing zero if needed: Write decimals with the same number of decimal places. For example, rewrite $0.5$ as $0.50$ to compare it with $0.48$.
   3. Compare digits from left to right: Looking at the tenths place, 5 tenths is greater than 4 tenths, so $0.50 > 0.48$, which means $0.5 > 0.48$.

   Expressed as fractions: $\frac{50}{100} > \frac{48}{100}$.