Ex 3.5, 1

Transcript

Ex 3.5, 1 Without performing long division, determine which of the following rational numbers will have terminating decimals and which will be repeating: 7/20, 4/15 and 13/250. Then check your answers by explicitly performing the long divisions and expressing these rational numbers as decimals. A rational number 𝑝/𝑞 to be terminating decimal We check the prime factorisation of denominator If Denominator = q = 2n 5m where n ≥ 0 & m ≥ 0 Then, our fraction is terminating Let’s check our fractions For 𝟕/𝟐𝟎 Doing prime factorisation of denominator 20 = 2 × 2 × 5 = 22 × 5 Since it only contains 2s and 5s, it will be a terminating decimal Let’s check long division Thus, 𝟕/𝟐𝟎 = 0.35 So, it terminates For 𝟒/𝟏𝟓 Doing prime factorisation of denominator 15 = 3 × 5 Because there is a 3 present, it will be a repeating decimal. Let’s check long division Thus, 𝟒/𝟏𝟓 = 0.2666… = 𝟎.𝟐𝟔 ̅ So, it is a non-terminating, repeating expansion For 𝟏𝟑/𝟐𝟓𝟎 Doing prime factorisation of denominator 250 = 2 × 5 × 5 × 5 = 2 × 53 Since it only contains 2s and 5s, it will be a terminating decimal Let’s check long division Thus, 𝟏𝟑/𝟐𝟓𝟎 = 0.052 So, it terminates