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Convert Shaded Circle Quantities to Mixed Number and Improper Fraction (Simplest Form)
Mathematics
Grade 5 (Primary School)
Question Content
Write the total number of shaded circles below as a) a mixed number. b) an improper fraction. Give each answer in its simplest form. (There are two circles, each divided into 8 equal parts; the first circle is fully shaded, the second has 6 parts shaded.)
Correct Answer
a) $1\\frac{3}{4}$; b) $\\frac{7}{4}$
Detailed Solution Steps
1
Step 1: Analyze the shaded parts of each circle. The first circle is fully shaded, so it represents 1 whole. The second circle is divided into 8 equal parts, with 6 parts shaded, so it represents $\\frac{6}{8}$.
2
Step 2: Simplify $\\frac{6}{8}$ by dividing both the numerator and denominator by their greatest common divisor (2), getting $\\frac{3}{4}$. For part a), combine the whole circle and the simplified fraction of the second circle: $1 + \\frac{3}{4} = 1\\frac{3}{4}$.
3
Step 3: For part b), convert the mixed number to an improper fraction. Multiply the whole number (1) by the denominator of the fraction (4), then add the numerator (3): $(1×4)+3=7$. Keep the denominator the same, so the improper fraction is $\\frac{7}{4}$. Alternatively, calculate total shaded parts: first circle has 8 shaded parts, second has 6, total 14. Total parts per circle is 8, so $\\frac{14}{8} = \\frac{7}{4}$ after simplifying.
Knowledge Points Involved
1
Fraction Simplification
This is the process of reducing a fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). For example, $\\frac{6}{8}$ simplifies to $\\frac{3}{4}$ because the GCD of 6 and 8 is 2. It is used to present fractions in their most concise form, making calculations and comparisons easier.
2
Mixed Numbers
A mixed number is a combination of a whole number and a proper fraction, used to represent quantities that are more than one whole but not a whole number. For example, $1\\frac{3}{4}$ means 1 whole plus $\\frac{3}{4}$ of another whole. It is commonly used in real-life scenarios to describe measurements or counts that include partial amounts.
3
Improper Fractions
An improper fraction is a fraction where the numerator is greater than or equal to the denominator, representing a quantity that is at least one whole. To convert a mixed number to an improper fraction, multiply the whole number by the denominator, add the numerator, and place the result over the original denominator. Improper fractions are useful for arithmetic operations like addition, subtraction, multiplication, and division of fractions.
4
Fraction Representation of Whole and Partial Quantities
This involves using fractions to describe parts of a whole. When a shape is divided into equal parts, each part is a unit fraction, and the number of shaded parts represents the numerator, while the total number of parts is the denominator. This is foundational for understanding how fractions relate to real-world visual representations.
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