Evaluate one and one-sixth divided by three-quarters multiplied by negative four-thirds divided by three, giving the answer in its simplest form.
Let’s begin by looking at the first parentheses or bracket, one and one-sixth divided by three-quarters. Our first step here is to convert one and one-sixth into an improper or top-heavy fraction. We do this by multiplying the whole number by the denominator and then adding the numerator. This gives us seven. Therefore, one and one-sixth is the same as seven-sixths. We need to divide this by three-quarters. When dividing two fractions, we can use the acronym KCF. We keep the first fraction the same, in this case, seven over six. We change the division sign to a multiplication sign. We flip or find the reciprocal of the second fraction, in this case, four over three.
Before multiplying these two fractions, we notice that we can cross simplify or cross cancel by dividing the four and six by two. Multiplying the numerators at this stage gives us 14. And multiplying the denominators gives us nine. One and one-sixth divided by three-quarters is 14 over nine or fourteen-ninths.
Let’s now consider our second parentheses or bracket. We need to divide negative four-thirds by three. As any integer can be written as this number over one, we can rewrite this as negative four-thirds divided by three over one. We now need to repeat the process from the first parentheses. This gives us negative four over three multiplied by one over three or one-third. Multiplying the numerators gives us negative four, and multiplying the denominators gives us nine. Recalling that multiplying a negative number by a positive number gives a negative answer and also that we can let the numerator or denominator of a negative fraction be negative.
Alternatively, we could consider the whole fraction as negative. Negative four-thirds divided by three is negative four-ninths. The two parentheses have therefore simplified to 14 over nine and negative four over nine. Once again, we need to multiply the numerators and then multiply the denominators. 14 multiplied by negative four is negative 56. Nine multiplied by nine is 81. The answer written as a fraction in its simplest form is negative 56 over 81.