# If a + b : b + c : c + a = 2 : 3 : 5 and a + b + c

= 20, then the value of c is:

# à¤¯à¤¦ि a + b : b + c : c + a = 2 : 3 : 5 à¤”à¤° a + b +

c = 20 à¤¹ै, à¤¤ो c à¤•ा à¤®ाà¤¨ à¤¹ै:

# (a) 12 (b) 5

(c) 10 (d) 2

SSC GD 2022 TIER-1

### Solution :

**Set up the Proportions:**

We have the proportions:

a + b : b + c : c + a = 2 : 3 : 5

**Solve for a, b, and c:**

Let's represent the common factor in these proportions as 'x'. So we have:

a + b = 2x

b + c = 3x

c + a = 5x

We also know a + b + c = 20.

Let's substitute 'a + b' with '2x' and 'b + c' with '3x' in the equation a + b + c = 20:

2x + 3x = 20

5x = 20

x = 4

Now we can find the values of a, b, and c:

a + b = 2x = 8

b + c = 3x = 12

c + a = 5x = 20

Solving these simultaneously, we get:

a = 4

b = 4

c = 12

**Answer:** The value of c is **12**. So the correct answer is (a).

*Speed Math Solution (with explanation) : *

**Given:
$\ufffd+\ufffd:\ufffd+\ufffd:\ufffd+\ufffd=2:3:5$
$\ufffd+\ufffd+\ufffd=20$**

**We quickly find that $\ufffd=4$.**

**Then, $\ufffd=5\ufffd-(\ufffd+\ufffd)=5(4)-(2\ufffd)=20-8=12$.**

**So, the value of $\ufffd$ is 12.**

** **

Given that $\ufffd+\ufffd+\ufffd=20$ and the ratios $\ufffd+\ufffd:\ufffd+\ufffd:\ufffd+\ufffd=2:3:5$,

you can observe that $\ufffd+\ufffd$ must be the largest sum because it has the highest ratio. So, $\ufffd+\ufffd=5\ufffd$, where $\ufffd$ is the common multiplier.

Since $\ufffd+\ufffd=20$, and the largest sum $\ufffd+\ufffd$ corresponds to 5x, which means $5\ufffd=20$, you can quickly deduce $\ufffd=4$.

Now, you directly know that $\ufffd=5\ufffd-\ufffd=5(4)-8=20-8=12$.

Therefore, you can determine that the value of $\ufffd$ is indeed 12 without explicitly calculating $\ufffd$ and $\ufffd$. This method allows you to solve the problem rapidly using mental math.