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Number - Fractions, Decimals, and Percentages

Grade 7IB

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

🔑Concepts

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Equivalence between Fractions, Decimals, and Percentages: Every rational number can be expressed in three ways. For example, 1/21/2 is the same as 0.50.5 or 50%50\%. Visually, imagine a 10x10 grid (100 squares total); if you shade 50 squares, you are highlighting 50100\frac{50}{100} of the area, which represents the decimal 0.50.5 and the percentage 50%50\%.

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Simplifying Fractions: A fraction is in its simplest form when the numerator and denominator have no common factors other than 1. This is done by dividing both by their Greatest Common Factor (GCF). Visually, 24\frac{2}{4} and 12\frac{1}{2} represent the same 'slice' of a circular pie, but 12\frac{1}{2} uses the fewest possible pieces to describe that area.

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Ordering and Comparing: To compare different formats, convert them all to decimals. On a horizontal number line, numbers further to the right are larger. For instance, to compare 34\frac{3}{4}, 0.80.8, and 70%70\%, convert them to 0.750.75, 0.800.80, and 0.700.70 respectively to see that 0.80.8 is the greatest.

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Operations with Decimals: When adding or subtracting decimals, you must align the decimal points vertically to ensure you are adding digits of the same place value (tenths to tenths, hundredths to hundredths). In multiplication, the number of decimal places in the product is the sum of the decimal places in the factors.

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Percentage Increase and Decrease: This measures how much a value has changed relative to its starting amount. Visually, a percentage increase can be seen as a 'bar model' where the original bar represents 100%100\% and an additional smaller bar is tacked onto the end to show the growth.

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Mixed Numbers and Improper Fractions: An improper fraction like 54\frac{5}{4} has a numerator larger than its denominator. This can be visualized as one whole circle plus one-quarter of another circle, which is written as the mixed number 1141 \frac{1}{4}.

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Terminating and Recurring Decimals: Some fractions like 14\frac{1}{4} result in terminating decimals (0.250.25), while others like 13\frac{1}{3} result in recurring decimals (0.333...0.333... or 0.3ˉ0.\bar{3}). On a calculator, a recurring decimal shows a repeating pattern of digits forever.

📐Formulae

Percentage=PartWhole×100\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100

Percentage Change=New Value−Original ValueOriginal Value×100\text{Percentage Change} = \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \times 100

Fraction to Decimal=Numerator÷Denominator\text{Fraction to Decimal} = \text{Numerator} \div \text{Denominator}

Decimal to Percentage=Decimal Value×100\text{Decimal to Percentage} = \text{Decimal Value} \times 100

💡Examples

Problem 1:

Calculate the final price of a jacket that originally costs 60aftera60 after a 25%$ discount.

Solution:

Step 1: Find the discount amount by calculating 25%25\% of 6060. 25% of 60=0.25×60=1525\% \text{ of } 60 = 0.25 \times 60 = 15 Step 2: Subtract the discount from the original price. 60−15=4560 - 15 = 45 The final price is $45.

Explanation:

To find the discount, we convert the percentage to a decimal and multiply by the original price, then subtract that value from the total.

Problem 2:

Solve 25+13\frac{2}{5} + \frac{1}{3} and provide the answer in simplest form.

Solution:

Step 1: Find a common denominator for 5 and 3, which is 15. Step 2: Convert the fractions. 25=2×35×3=615\frac{2}{5} = \frac{2 \times 3}{5 \times 3} = \frac{6}{15} 13=1×53×5=515\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15} Step 3: Add the numerators. 615+515=1115\frac{6}{15} + \frac{5}{15} = \frac{11}{15}

Explanation:

Fractions can only be added when they have the same denominator. We use the Lowest Common Multiple (LCM) to find a common denominator before adding the numerators.