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 :
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