Video: Evaluating Numerical Expressions Involving Recurring Decimals and Mixed Numbers

Evaluate 1 1/6 + (−0.06) recurring giving the answer in its simplest form.

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Video Transcript

Evaluate one and one-sixth plus negative 0.06 recurring giving the answer in its simplest form.

In order to answer this question, we need to write both terms in their simplest form as fractions. The fraction one and one-sixth can be written as an improper or top-heavy fraction by multiplying the whole number by the denominator and then adding the numerator. One multiplied by six is equal to six. Adding one to this gives us seven. This means that the fraction one and one-sixth is equal to seven-sixths. We could also show this pictorially by splitting two rectangles into six pieces. Shading one whole rectangle and one-sixth of the second rectangle is the same as seven-sixths altogether.

We also need to convert negative 0.06 recurring into a fraction. Let’s first consider the decimal 0.06 recurring. When there are two digits after the decimal point that both repeat, we can convert the value to a fraction by writing the repeating part as the numerator and 99 as the denominator. This means that the decimal 0.06 recurring is the same a six over 99. In the same way, the decimal 0.23 recurring is equal to 23 out of 99. If there is one digit after the decimal point that is repeating, we write this over nine. 0.7 recurring is equal to seven-ninths. The decimal 0.128 recurring is equal to 128 over 999. As long as all the digits after the decimal point are repeating, this pattern will hold.

We can simplify the fraction six over 99 by dividing the numerator and denominator by three. Six divided by three is equal to two. And 99 divided by three is equal to 33. Therefore, the decimal 0.06 recurring simplifies to two over 33. The original question one and one-sixth plus negative 0.06 recurring can therefore be rewritten as seven-sixths plus negative two thirty-thirds. Adding a negative number is the same as subtracting the number. We need to calculate seven-sixths minus two thirty-thirds. In order to add or subtract two fractions, we need to find the lowest common denominator.

In this case, this will be the lowest common multiple of six and 33. This is equal to 66. There are 11 sixes in 66. So we need to multiply the top and bottom of the first fraction by 11. This is equal to 77 over 66. 33 multiplied by two is equal to 66. So we need to multiply the top and bottom of the second fraction by two. Two multiplied by two is equal to four. So we’re left with four over 66. As the denominators are now the same, we can just subtract the numerators. 77 minus four is equal to 73. Seven-sixths minus two thirty-thirds is equal to seventy-three sixty-sixths.

The calculation one and one-sixth plus negative 0.06 recurring, in its simplest form, is 73 over 66. As 73 and 66 have no common factor, this fraction can’t be simplified. We could, however, write it as a mixed fraction, one and seven over 66. Either of these answers is correct.

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