Explanation

To find the fourth proportional of 3, 7, and 15:

Let's denote the fourth proportional as x. We set up the proportion:

3/7 = 15/x

Step 1: Cross-multiply.

3x = 7 × 15

Step 2: Calculate the product of 7 and 15.

3x = 105

Step 3: Solve for x.

x = 105 / 3

x = 35

Explanation

Given CF = 24 and the centroid G divides CF in a 2:1 ratio, with the longer part toward the vertex C and the shorter part toward the midpoint F of side AB.

The total length CF is divided into 3 parts, where 2 parts are toward the vertex (CG) and 1 part is toward the midpoint (FG).

FG = (1/3) × CF = (1/3) × 24 = 8.

Explanation

Given:

105 increased in the ratio 9 : 7

To increase 105 in the ratio 9 : 7, we'll multiply 105 by 9/7:

New value = 105 × (9/7)

= 15 × 9

= 135

Explanation

To find the fourth proportional, let's denote it as x. The proportion will be:

3 : 9 = 7 : x

We can cross-multiply:

3x = 9 × 7

3x = 63

x = 63 / 3

x = 21

Explanation

Given:

g : h = 8 : 9

h : i = 9 : 12

To find the combined ratio g : h : i, we need to make the ratio of h the same in both.

g : h = 8 : 9

h : i = 9 : 12

Since h has the same ratio in both (9), we can directly combine:

g : h : i = 8 : 9 : 12