If $$ \frac{a}{b+c}=\frac{1}{3}, \frac{c}{a+b}=\frac{5}{7} $$, Find the value of: $$ \frac{b}{a+c} $$

Question

UpStudy AI · Accepted Answer

We are given the following equations:

1. $$\frac{a}{b + c} = \frac{1}{3}$$
2. $$\frac{c}{a + b} = \frac{5}{7}$$

Our goal is to find the value of $$\frac{b}{a + c}$$.

**Step 1: Express $$a$$ and $$c$$ in terms of $$b$$**

From equation (1):
$$
a = \frac{1}{3}(b + c)
$$

From equation (2):
$$
c = \frac{5}{7}(a + b)
$$

**Step 2: Substitute $$a$$ from equation (1) into equation (2)**

$$
c = \frac{5}{7}\left(\frac{1}{3}(b + c) + b\right)
$$
$$
c = \frac{5}{7}\left(\frac{b + c + 3b}{3}\right)
$$
$$
c = \frac{5}{7} \times \frac{4b + c}{3}
$$
$$
21c = 5(4b + c)
$$
$$
21c = 20b + 5c
$$
$$
16c = 20b
$$
$$
c = \frac{5}{4}b
$$

**Step 3: Find $$a$$ in terms of $$b$$**

Using $$c = \frac{5}{4}b$$ in equation (1):
$$
a = \frac{1}{3}\left(b + \frac{5}{4}b\right) = \frac{1}{3} \times \frac{9}{4}b = \frac{3}{4}b
$$

**Step 4: Calculate $$\frac{b}{a + c}$$**

$$
a + c = \frac{3}{4}b + \frac{5}{4}b = 2b
$$
$$
\frac{b}{a + c} = \frac{b}{2b} = \frac{1}{2}
$$

**Answer:** $$\boxed{\dfrac{1}{2}}$$
$$\frac{b}{a + c} = \frac{1}{2}$$

If , Find the value of:

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions