Question 35 of 58beginner💡 UnderstandNumerical3 marks

Try to find the decimal expansions of 103\frac{10}{3} and 1112\frac{11}{12}. What do you observe about the repetition of the digits after the decimal point?

Correct Answer

Decimal expansion of 103\frac{10}{3}: When we divide 10 by 3, we get 3.333... The digit '3' repeats infinitely. So, 103=3.3‾\frac{10}{3} = 3.\overline{3}.

Decimal expansion of 1112\frac{11}{12}: When we divide 11 by 12, we get 0.91666... The digit '6' repeats infinitely. So, 1112=0.916‾\frac{11}{12} = 0.91\overline{6}.

Observation: In both cases, the decimal expansions do not terminate. Instead, a sequence of digits begins to loop infinitely after the decimal point. This happens because during the long division, the remainders start repeating, which causes the digits in the quotient to repeat as well, creating a repeating decimal.

Exercise: Think and Reflect | Q: 1 | (Page 17)
For More Understanding

Explanation

The textbook context explains that for some rational numbers, the division never reaches a remainder of 0. Because there is a limited number of possible remainders, eventually a remainder must show up a second time. Once a remainder repeats, the entire division process loops, creating a repeating decimal. In 10/3, the remainder is always 1, causing 3 to repeat. In 11/12, the remainder eventually becomes 8 repeatedly, causing 6 to repeat.

Solution Steps

  1. Step 1: Divide 10 by 3 to get 3.333..., written as 3.3‾3.\overline{3}.

  2. Step 2: Divide 11 by 12 to get 0.91666..., written as 0.916‾0.91\overline{6}.

  3. Step 3: Observe that both expansions have digits that repeat infinitely after the decimal point.

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