# Four-fifths of my current age is greater than three-quarters of my age one year from now. Four-fifths of my current age is also greater than five-sixths of my age one year ago. Given that my age is an integer, what are all possible values for my age? Explain how you determined your answer, showing all steps along the way.

###### Question:

Four-fifths of my current age is also greater than five-sixths of my age one year ago.

Given that my age is an integer, what are all possible values for my age?

Explain how you determined your answer, showing all steps along the way.

## Answers

let current age be x, next we construct the inequalities for the age:Four-fifths of my current age is greater than three-quarters of my age one year from now.thus when we add 1 year to 3/4 of our current age and set the inequality4x/5>3x/4+1......iFour-fifths of my current age is also greater than five-sixths of my age one year ago.Thus when we subtract 1 from 5x/6 and set the inequality we get4x/5>5x/6-1.......iisolving the inequalities we obtain:4x/5>3x/4+1x/20>1hence multiplying both sides by 20 we obtain:x>20also4x/5>5x/6-14x/5-5x/5>-1-x/30>-1multiplying both sides by 30 we get:-x>-30thusx<30therefore the possible values of my age will lie in the interval:20<x<30Thus our age is in the interval(20,30)

Answer:The possible values of your age are 16, 17, 18, 19, 20, 21, 22, 23, and 24.Step-by-step explanation:Let a be current age.The first equation tells us that:4/5a > 3/4(a+1).First, we expand the parenthesis by multiplying 3/4 by a+1.4/5a > 3/4a +3/4.Now we move all like terms to their own side of the inequality. All terms with a variable like 'a' will go to the left and all therms without a variable will go to the right.4/5a -3/4a > 3/4.We can now find a common denominator. LCM(5,4) = 20.16/20a - 15/20a > 15/20.We can now subtract terms with a variable.1/20a > 15/20.We can multiply each fraction by 20 to get rid of the denominators.20a > 300Now, we divide by 20 to get a alone.a > 15The current age has to be greater than 15.The second equation tells us that:4/5a > 5/6(a-1).To solve, we isolate a. First, we expand the parenthesis by multiplying 5/6 by a-1.4/5a > 5/6a - 5/6.Next, we move all like terms to one side of the inequality. We'll move all the terms with 'a' in it to the left side and all the terms without 'a' to the right side.4/5a - 5/6a > -5/6.We can find a common denominator between all the fractions. The least common multiple of 5 and 6 is 30.24/30a - 25/30a > -25/30.Now, we can subtract like terms.-1/30a > -25/30.We multiply both sides by 30 to get rid of the denominators.-30a > -750.We divided both side by -30. We make sure to switch the direction of the sign because we divided by a negative number.a < 25.The current age has to be less than 25.We know that a > 15 and a < 25. A better way of writing this would be:15 < a < 25. This tells us a has to be greater than 15 but less than 25. The integers between 15 and 25 are 16, 17, 18, 19, 20, 21, 22, 23, and 24.