Question

Add or Subtract Fraction.

$$\begin{gathered} \overline{)a}\frac{5n}{6}÷\frac{8}{15} \\ \overline{)b}\frac{5n}{6}-\frac{8}{15} \end{gathered}$$

Short answer

Part (a) The value of $\frac{5n}{6}÷\frac{8}{15}$is $\frac{25n}{12}$.

Part (b) The value oflocalid="1645593048420" $\frac{5n}{6}-\frac{8}{15}$is $\begin{array}{rcl}& & \frac{25n-16}{30}\end{array}$.

Step-by-step solution

Part (a) Step 1. Given information

Consider the expression.

$\overline{)a}\frac{5n}{6}÷\frac{8}{15}$

Part (a)  Step 2. Determine the operation.

The operation is division.

Part (a)  Step 3. Apply the Fraction Division Formula:ab÷cd=ab·dc, where b≠0, c≠0, and src="data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHhtbG5zOndycz0iaHR0cDovL3d3dy53aXJpcy5jb20veG1sL21hdGhtbC1leHRlbnNpb24iIGhlaWdodD0iMjAiIHdpZHRoPSIzNyIgd3JzOmJhc2VsaW5lPSIxNiI%2BPCEtLU1hdGhNTDogPG1hdGggeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzE5OTgvTWF0aC9NYXRoTUwiPjxtaT5kPC9taT48bW8%2BJiN4MjI2MDs8L21vPjxtbj4wPC9tbj48L21hdGg%2BLS0%2BPGRlZnM%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%2BPHRleHQgZm9udC1mYW1pbHk9IkFyaWFsIiBmb250LXNpemU9IjE2IiBmb250LXN0eWxlPSJpdGFsaWMiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjQuNSIgeT0iMTYiPmQ8L3RleHQ%2BPHRleHQgZm9udC1mYW1pbHk9Im1hdGgxN2FjNTJlM2YyNzI5ZDFiM2Y2ZDJiN2U4ZjYiIGZvbnQtc2l6ZT0iMTYiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjE4LjUiIHk9IjE2Ij4mI3gyMjYwOzwvdGV4dD48dGV4dCBmb250LWZhbWlseT0iQXJpYWwiIGZvbnQtc2l6ZT0iMTYiIHRleHQtYW5jaG9yPSJtaWRkbGUiIHg9IjMxLjUiIHk9IjE2Ij4wPC90ZXh0Pjwvc3ZnPg%3D%3D" localid="1644393668653" src="https://studysmarter-mediafiles.s3.amazonaws.com/media/textbook-exercise-images/97e6a51c-99b5-46fc-9cb2-eeda6f3b3f3f.svg?X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIA4OLDUDE42UZHAIET%2F20220225%2Feu-central-1%2Fs3%2Faws4_request&X-Amz-Date=20220225T102407Z&X-Amz-Expires=90000&X-Amz-SignedHeaders=host&X-Amz-Signature=0d937c9daa98386f2087b694cdb5f383d8c46fd48d73691ded3c7aa0b879d221" d≠0.

$\begin{array}{rcl}\frac{5n}{6}÷\frac{8}{15}& =& \frac{5n}{6}·\frac{15}{6}\\ & =& \frac{5n}{\cancel{3}·2}·\frac{\cancel{3}·5}{6}\\ & =& \frac{25n}{12}\\ & & \end{array}$

Thus, the value is $\frac{25n}{12}$.

Part (b) Step 1. Now, consider the expression.b5n6-815 Determine the operation.

The operation is subtraction.

Part (b) Step 2. Determine the whether the given fraction has the same denominator.

The given fraction does not have the same denominator.

So, find the least common denominator of $6$and $15$.

$\begin{array}{rcl}6& =& 2·3\\ 15& =& 3·5\\ & \Rightarrow & LCM=2·3·5\\ & \Rightarrow & LCM=30\end{array}$

Part (b) Step 3. Multiply the numerator and denominator of each fraction by a factor to get the denominator 30.

$\begin{array}{rcl}\frac{5n}{6}-\frac{8}{15}& =& \frac{5n·5}{6·5}-\frac{8·2}{15·2}\\ & =& \frac{25n}{30}-\frac{16}{30}\end{array}$

Part (b) Step 4. Apply the Fraction Subtraction rule:ac-bc=a-bc, where c≠0.

$\begin{array}{rcl}\frac{5n}{6}-\frac{8}{15}& =& \frac{25n-16}{30}\end{array}$

Hence, the value is $\begin{array}{rcl}& & \frac{25n-16}{30}\end{array}$.