Problem 1 Consider the following pairs of positive numbers with specific relati

Problem 1

Consider the following pairs of positive numbers with specific relationships between their sums and differences. Solve for the given ratios in each case:

a.

Two positive numbers a and b, with a > b, are such that three times their sum is equal to four times their difference. What is the ratio a : b?

b.

Two positive numbers x and y, with x > y, are such that four times their sum is equal to five times their difference. What is the ratio x : y?

c.

Two positive numbers p and q, with p > q, are such that the sum of three times p and twice q is equal to five times their difference. What is the ratio p : q?

d.

Two positive numbers m and n, with m > n, are such that twice their sum is equal to five times their difference. What is the ratio m : n?

e.

Two positive numbers u and v, with u > v, are such that three times their sum is equal to six times their difference. What is the ratio u : v?

Note: Use algebraic methods to solve each of these ratio problems, by setting up appropriate equations and finding the correct relationship between the numbers.

Geometry Problems Involving Ratios

Problem 2

In triangle ABC, the ratio of angle A to angle B to angle C is 2 : 3 :