QuestionCombine $\frac{l}{c}+\frac{l}{d}-\frac{c-d}{c d}$ into one simplified fraction.

Studdy Solution

STEP 1

Assumptions1. The given expression is $\frac{l}{c}+\frac{l}{d}-\frac{c-d}{c d}$ . . We need to simplify this expression to a single fraction.

STEP 2

First, we notice that the first two terms have a common numerator. We can combine these two terms into a single fraction. The common denominator will be the product of the two denominators.
$\frac{l}{c}+\frac{l}{d} = \frac{l \cdot d + l \cdot c}{c \cdot d}$

STEP 3

Next, we simplify the numerator of the new fraction by factoring out the common factor, which is $l$ .
$\frac{l \cdot d + l \cdot c}{c \cdot d} = \frac{l(c + d)}{c \cdot d}$

STEP 4

Now, we rewrite the entire expression, replacing the first two terms with the new fraction we just calculated.
$\frac{l(c + d)}{c \cdot d} - \frac{c-d}{c \cdot d}$

STEP 5

We notice that the two fractions have the same denominator. This means we can combine them into a single fraction.
$\frac{l(c + d)}{c \cdot d} - \frac{c-d}{c \cdot d} = \frac{l(c + d) - (c - d)}{c \cdot d}$

STEP 6

Next, we simplify the numerator of the new fraction.
$\frac{l(c + d) - (c - d)}{c \cdot d} = \frac{lc + ld - c + d}{c \cdot d}$

STEP 7

Now, we group the terms in the numerator.
$\frac{lc + ld - c + d}{c \cdot d} = \frac{lc - c + ld + d}{c \cdot d}$

STEP 8

Next, we factor out the common factors from the grouped terms.
$\frac{lc - c + ld + d}{c \cdot d} = \frac{c(l -1) + d(l +1)}{c \cdot d}$ So, the expression $\frac{l}{c}+\frac{l}{d}-\frac{c-d}{c d}$ simplifies to $\frac{c(l -1) + d(l +1)}{c \cdot d}$ .

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