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Decimals - Conversion of Fractions to Decimals and vice-versa

Grade 5ICSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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Place Value of Decimals: A decimal number has two parts: the whole number part and the decimal part, separated by a dot called the decimal point. To the left of the point, places are ones, tens, hundreds, etc. To the right, places are tenths (110\frac{1}{10}), hundredths (1100\frac{1}{100}), and thousandths (11000\frac{1}{1000}). Imagine a place value chart where the decimal point acts as a central divider between whole units and fractional parts.

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Converting Fractions with Denominators 10, 100, 1000: To convert a fraction like 710\frac{7}{10} or 23100\frac{23}{100} to a decimal, count the number of zeros in the denominator and move the decimal point in the numerator to the left by the same number of places. For example, in 9100\frac{9}{100}, there are two zeros, so we move the point two places left to get 0.090.09. Visually, if you have a grid of 100 small squares and shade 9, you have 0.090.09 of the whole.

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Converting Fractions by Division: Any fraction can be converted to a decimal by dividing the numerator by the denominator. If the division is not exact, add a decimal point and extra zeros to the dividend (numerator) to continue dividing. For example, to convert 34\frac{3}{4}, you divide 3.003.00 by 44 to get 0.750.75.

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Converting Fractions using Equivalent Fractions: Fractions with denominators like 2,4,5,20,25,502, 4, 5, 20, 25, 50 can be converted by multiplying both the numerator and denominator by a number that makes the denominator 10,100,or 100010, 100, \text{or } 1000. For instance, 15=1×25×2=210=0.2\frac{1}{5} = \frac{1 \times 2}{5 \times 2} = \frac{2}{10} = 0.2.

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Converting Mixed Numbers to Decimals: A mixed number like 4124 \frac{1}{2} consists of a whole number 44 and a fraction 12\frac{1}{2}. Keep the whole number to the left of the decimal point and convert only the fractional part. Since 12=0.5\frac{1}{2} = 0.5, the decimal form is 4.54.5.

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Converting Decimals to Fractions: To convert a decimal to a fraction, write the decimal number (without the point) as the numerator. For the denominator, write 11 followed by as many zeros as there are decimal places. For example, 0.150.15 has two decimal places, so it becomes 15100\frac{15}{100}.

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Simplifying the Fraction: After converting a decimal to a fraction, always reduce it to its simplest form by dividing the numerator and denominator by their Highest Common Factor (HCF). For 15100\frac{15}{100}, divide both by 55 to get 320\frac{3}{20}.

📐Formulae

Decimal Place Value=⋯+(Tens×10)+(Ones×1)+(Tenths×110)+(Hundredths×1100)+…\text{Decimal Place Value} = \dots + (\text{Tens} \times 10) + (\text{Ones} \times 1) + (\text{Tenths} \times \frac{1}{10}) + (\text{Hundredths} \times \frac{1}{100}) + \dots

Decimal=NumeratorDenominator\text{Decimal} = \frac{\text{Numerator}}{\text{Denominator}}

Fraction=Number without decimal point10n (where n is the number of decimal places)\text{Fraction} = \frac{\text{Number without decimal point}}{10^{n}} \text{ (where } n \text{ is the number of decimal places)}

💡Examples

Problem 1:

Convert the fraction 725\frac{7}{25} into a decimal.

Solution:

Step 1: Find a number to multiply the denominator to make it 100. Since 25×4=10025 \times 4 = 100, we multiply both numerator and denominator by 4. \ 7×425×4=28100\frac{7 \times 4}{25 \times 4} = \frac{28}{100} \ Step 2: Since there are two zeros in 100, move the decimal point in 28 two places to the left. \ 28100=0.28\frac{28}{100} = 0.28

Explanation:

To convert a fraction to a decimal, it is often easiest to make the denominator a power of 10 (like 10, 100, or 1000) using equivalent fractions.

Problem 2:

Convert the decimal 0.3750.375 into a fraction in its simplest form.

Solution:

Step 1: Count the decimal places. There are 3 decimal places (3, 7, and 5). \ Step 2: Write the decimal as a fraction with 10001000 (one followed by three zeros) as the denominator. \ 0.375=37510000.375 = \frac{375}{1000} \ Step 3: Simplify the fraction by dividing both numerator and denominator by their common factors. Dividing by 25: \ 375÷251000÷25=1540\frac{375 \div 25}{1000 \div 25} = \frac{15}{40} \ Step 4: Divide by 5 again to reach simplest form: \ 15÷540÷5=38\frac{15 \div 5}{40 \div 5} = \frac{3}{8}

Explanation:

First, express the decimal as a fraction using the place value of the last digit (thousandths), then divide by common factors until the fraction cannot be simplified further.