Questions: Relating Fractions and Decimals

Question 1

A student claims that 1/3 = 0.3 because 'there's a 3 in both.' What is wrong with this reasoning?

• A. Nothing — 1/3 and 0.3 are the same number and represent the same point on the number line
• B. 0.3 means 3/10, not 1/3 — the numbers look similar but represent very different quantities
• C. 1/3 should be written as 0.03, not 0.3
• D. The student should divide 1 ÷ 3 to get the decimal, which equals 3.0

Answer: (B)

0.3 means 3 tenths, or 3/10 — not 1 third. 3/10 = 0.3 exactly, while 1/3 ≈ 0.333... (a repeating decimal that never ends). The confusion comes from surface similarity: both involve the digit 3. The actual conversion from fraction to decimal goes through the powers-of-ten structure. 1/3 cannot be written as a clean decimal because 3 does not divide evenly into 10 or 100.

Question 2

To convert 3/4 to a decimal, a student finds that 4 × 25 = 100 and multiplies both numerator and denominator by 25, getting 75/100. What decimal does 75/100 equal?

• A. 0.075 — because 75 is placed three places after the decimal point
• B. 0.75 — because 75/100 means 75 hundredths
• C. 7.5 — because 75 divided by 10
• D. 0.34 — because the fraction converts digit by digit (3 and 4)

Answer: (B)

75/100 reads as '75 hundredths,' which is 0.75. The strategy works because the decimal system is built on powers of 10: hundredths place = /100. Since 4 × 25 = 100 and 3 × 25 = 75, we get 75/100 = 0.75. This is the benchmark ¾ = 0.75, which should become automatic.

Question 3

The fraction 7/10 and the decimal 0.7 name exactly the same quantity.

• T. True
• F. False

Answer: True

Decimals with one digit after the decimal point represent tenths. 0.7 literally means '7 tenths,' which is exactly what 7/10 means. They are two different notations for the same number — like writing 'a dozen' and '12.' Place them on a number line and they land on the exact same point.

Question 4

Fractions represent parts of a whole, while decimals represent measurements — they are fundamentally different types of numbers that happen to look similar sometimes.

• T. True
• F. False

Answer: False

Fractions and decimals are the same numbers expressed in different notations, not different types of numbers. 1/2 and 0.5 occupy exactly the same point on the number line. The decimal system is built on fractions with powers-of-ten denominators; 0.7 is simply shorthand for 7/10. Treating them as separate categories is the central misconception this topic is designed to correct.

Question 5

Why can 1/4 be expressed as a clean decimal (0.25), but 1/3 cannot?

Model answer: 1/4 can be converted to an equivalent fraction with denominator 100 (since 4 × 25 = 100), giving 25/100 = 0.25. But 3 does not divide evenly into 10 or 100 — there is no whole number you can multiply 3 by to get 10 or 100 — so 1/3 produces a repeating decimal (0.333...) rather than a clean one.

Decimals are built on powers of 10. A fraction converts cleanly to a decimal only when its denominator can be made into 10 or 100 (or another power of 10) by multiplying by a whole number. 4 × 25 = 100, so 1/4 = 25/100 = 0.25. But 3 shares no factors with 10, so dividing 1 by 3 never terminates. This is why benchmark fractions like 1/4, 1/5, and 1/2 have clean decimal equivalents while 1/3, 1/6, and 1/7 do not.