Decimals and Fractions

Decimals and Fractions

A fraction names equal parts: the denominator gives the number of equal parts in a whole, and the numerator counts those parts. Decimal places name tenths, hundredths, and thousandths.

How to solve these problems

Compare values by rewriting them in the same form. Trailing decimal zeros do not change value. Equivalent fractions multiply or divide numerator and denominator by the same nonzero number.

Worked Example 1: Describe 3/4

Describe 3/4

Solution: The denominator 4 divides a whole into four equal pieces; the numerator 3 selects three pieces.

Why it works: Equal-sized parts are essential: three unequal pieces need not represent three-fourths.

Worked Example 2: Read 0.37

Read 0.37

Solution: 0.37 = 3/10 + 7/100 = 37/100.

Why it works: The first digit after the decimal counts tenths and the second counts hundredths.

Worked Example 3: Compare 0.5 and 0.50

Compare 0.5 and 0.50

Solution: 0.5 = 5/10 = 50/100 = 0.50.

Why it works: A trailing zero changes the written precision, not the numerical value.

Worked Example 4: Compare 0.4 and 0.35

Compare 0.4 and 0.35

Solution: Write 0.4 as 0.40. Since 40 hundredths > 35 hundredths, 0.4 > 0.35.

Why it works: Compare matching place values rather than the count of digits.

Worked Example 5: Simplify 6/8

Simplify 6/8

Solution: Divide numerator and denominator by 2: 6/8 = 3/4.

Why it works: Dividing both by the same nonzero factor preserves the ratio.

Worked Example 6: Find an equivalent fraction for 2/3 with denominator 12

Find an equivalent fraction for 2/3 with denominator 12

Solution: Multiply numerator and denominator by 4: 2/3 = 8/12.

Why it works: The scaling factor must apply to both parts.

Worked Example 7: Compare 1/2 and 0.6

Compare 1/2 and 0.6

Solution: 1/2 = 0.5, and 0.5 < 0.6.

Why it works: Using decimals puts both numbers on the same scale.

Worked Example 8: Order 0.25, 3/4, and 0.5

Order 0.25, 3/4, and 0.5

Solution: Rewrite 3/4 as 0.75. Increasing order: 0.25, 0.5, 3/4.

Why it works: Ordering requires comparing the values rather than the notation.

Worked Example 9: Write 7/4 as a mixed number

Write 7/4 as a mixed number

Solution: 7 ÷ 4 = 1 remainder 3, so 7/4 = 1¾.

Why it works: Four fourths make one whole, leaving three fourths.

Worked Example 10: Write 2⅓ as an improper fraction

Write 2⅓ as an improper fraction

Solution: 2 × 3 + 1 = 7, so 2⅓ = 7/3.

Why it works: Two wholes contain six thirds; adding one third gives seven thirds.

Try it again

Cover the solutions and solve the ten examples on your own. Explain which rule you used, then compare your method and answer with the worked solution. Check signs, equivalent forms, and units.