I have a problem solving the question below could someone help me solve it or at least give me a hint on how to solve it ? If you can help here is the question ; p,q,r are three consecutive terms of an A.P whose sum is 21. The ratio p:r = 6:1 , Find p:q:r.

Question

I have a problem solving the question below could someone help me solve it or at least give me a hint on how to solve it ? If you can help here is the question ; p,q,r are three consecutive terms of an A.P whose sum is 21. The ratio p:r = 6:1 , Find p:q:r.

Answer 1

p = x

q = x + d
r = x + 2d

x/(x+2d) = 6/1

3 x + 3 d = 21
so
x + d = 7
and
x = 6x + 12 d or 5 x = -12 d

5 x + 12 d = 0
5 x + 5 d = 35
-----------------
7 d = -35
d = -5
x = 12
etc

Answer 2

To solve this question, we need to use the given information about the sum and the ratio of the terms in the arithmetic progression (A.P.).

Let's start by assigning variables to the three consecutive terms: p, q, and r.

We know that the sum of the three terms is 21:

p + q + r = 21 ---(1)

We are also given that the ratio of p to r is 6:1. In other words, p is 6 times smaller than r:

p = (1/6) * r ---(2)

Now, we can substitute equation (2) into equation (1) to eliminate p:

(1/6) * r + q + r = 21

Combine like terms:

(7/6) * r + q = 21

Multiply both sides by 6 to get rid of the fraction:

7r + 6q = 126 ---(3)

Now, we have a system of two equations (equations 2 and 3) with two variables (q and r). We can solve this system to find the values of q and r.

One way to solve this system is to manipulate equation (2) and equation (3) to make q the subject of the formula, and then equate the two formulas together:

(1/6) * r = p
r = 6 * p

Substitute this value of r into equation (3):

7(6p) + 6q = 126

Simplify:

42p + 6q = 126

Divide by 6 to simplify further:

7p + q = 21 ---(4)

Now we have a new equation (4) that relates p and q. We can now solve this equation together with equation (2) to find the values of p, q, and r.

Substitute the value of p from equation (2) into equation (4):

7 * (1/6) * r + q = 21

Multiply through by 6 to get rid of the fraction:

7r + 6q = 126

Notice that this equation is the same as equation (3). This means that the values of q and r that satisfy equation (3) also satisfy equation (4).

Therefore, the solution to this system of equations is any pair of values for q and r that satisfy equation (3). In other words, there are infinitely many possible values for p, q, and r that satisfy the given conditions.

Hence, we cannot determine the exact values of p, q, and r or the ratio p:q:r without further information.