Decimals Cheatsheet - Class 6

Cheatsheet Content

1. Understanding Decimals Decimals are another way to write fractions with denominators of 10, 100, 1000, etc. Decimal Point: Separates the whole number part from the fractional part. Place Values: ...Hundreds, Tens, Ones . Tenths, Hundredths, Thousandths... Example: $12.345$ $1$ is in Tens place $2$ is in Ones place $3$ is in Tenths place ($\frac{3}{10}$) $4$ is in Hundredths place ($\frac{4}{100}$) $5$ is in Thousandths place ($\frac{5}{1000}$) 2. Converting Decimals to Fractions To convert a decimal to a fraction: Write the decimal as a numerator: Remove the decimal point. Determine the denominator: Count the number of digits after the decimal point. One digit after decimal $\rightarrow$ denominator is $10$ Two digits after decimal $\rightarrow$ denominator is $100$ Three digits after decimal $\rightarrow$ denominator is $1000$ And so on... Simplify the fraction: Divide the numerator and denominator by their greatest common factor (GCF). Examples: $0.5 = \frac{5}{10} = \frac{1}{2}$ $0.25 = \frac{25}{100} = \frac{1}{4}$ $1.75 = \frac{175}{100} = \frac{7}{4}$ (or $1\frac{3}{4}$) $0.123 = \frac{123}{1000}$ (cannot be simplified) 3. Converting Fractions to Decimals To convert a fraction to a decimal: Method 1: Make denominator a power of 10 (10, 100, 1000, etc.). Multiply both numerator and denominator by the same number to get $10, 100, 1000, \dots$ in the denominator. Then, write the numerator and place the decimal point according to the denominator. Method 2: Divide the numerator by the denominator. Use long division. Add zeros after the decimal point in the numerator if needed. Examples: $\frac{3}{5}$: Method 1: $\frac{3 \times 2}{5 \times 2} = \frac{6}{10} = 0.6$ Method 2: $3 \div 5 = 0.6$ $\frac{1}{4}$: Method 1: $\frac{1 \times 25}{4 \times 25} = \frac{25}{100} = 0.25$ Method 2: $1 \div 4 = 0.25$ $\frac{7}{8}$: Method 1: $\frac{7 \times 125}{8 \times 125} = \frac{875}{1000} = 0.875$ Method 2: $7 \div 8 = 0.875$ 4. Comparing Decimals To compare decimals: Compare whole number parts: The decimal with the larger whole number part is greater. If whole number parts are the same: Compare the digits in the tenths place. The one with the larger digit is greater. If tenths are the same: Compare the digits in the hundredths place, and so on. Tip: You can add trailing zeros to make the number of decimal places equal for easier comparison (e.g., $0.5$ is $0.50$). Examples: $3.45$ vs $2.98 \rightarrow 3.45 > 2.98$ (because $3 > 2$) $1.23$ vs $1.32 \rightarrow 1.32 > 1.23$ (because $3 > 2$ in tenths place) $0.75$ vs $0.705 \rightarrow 0.750$ vs $0.705 \rightarrow 0.75 > 0.705$ (because $5 > 0$ in hundredths place) 5. Comparing Fractions and Decimals To compare a fraction and a decimal, convert both to the same format: Option 1: Convert the fraction to a decimal. Then, compare the two decimals. Option 2: Convert the decimal to a fraction. Then, compare the two fractions (e.g., by finding a common denominator). Examples: Compare $\frac{3}{4}$ and $0.7$: Option 1 (Fraction to Decimal): $\frac{3}{4} = 0.75$. Now compare $0.75$ and $0.7$. Since $0.75 > 0.7$, then $\frac{3}{4} > 0.7$. Option 2 (Decimal to Fraction): $0.7 = \frac{7}{10}$. Now compare $\frac{3}{4}$ and $\frac{7}{10}$. Common denominator for $4$ and $10$ is $20$. $\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$ $\frac{7}{10} = \frac{7 \times 2}{10 \times 2} = \frac{14}{20}$ Since $\frac{15}{20} > \frac{14}{20}$, then $\frac{3}{4} > 0.7$. Compare $\frac{1}{2}$ and $0.45$: Convert $\frac{1}{2} = 0.5$. Compare $0.5$ and $0.45$. $0.5 > 0.45$, so $\frac{1}{2} > 0.45$.