Could you please explain the quickest way to calculate these questions?
1)Julian's age has the same figures as his dad's with the digits reversed. The sum of their ages is 77 and Julian is 27 years younger than his dad. How old is Julian?
2) Amanda has 20 coins in her purse. She has only 10c,20c and 50c coins, and their total value is $5. If she has more 50c than 10c coins, how many 10c has she?
Let D = dad's age
let J = Julian's age.
D+J=77
J+27=D
Two equations. Two unknowns. Solve for D and J.
Sure! Let's break down both questions and explain the quickest way to calculate the answers.
1) Julian's age has the same figures as his dad's with the digits reversed. The sum of their ages is 77 and Julian is 27 years younger than his dad. How old is Julian?
To solve this problem, we need to set up a system of equations. Let's denote Julian's age as 'J' and his dad's age as 'D'.
First, we know that Julian is 27 years younger than his dad, so we can write the equation: J = D - 27.
Second, we are told that Julian's age has the same figures as his dad's with the digits reversed. This means that the tens and units digits of their ages are swapped. We can express this as: J = 10u + t and D = 10t + u, where 'u' represents the units digit and 't' represents the tens digit.
Now we can set up a third equation based on the sum of their ages: J + D = 77.
To solve this system of equations, we can substitute J and D from the first two equations into the third equation:
(D - 27) + D = 77
2D - 27 = 77
2D = 104
D = 52
Substituting D = 52 back into the equation J = D - 27, we find:
J = 52 - 27
J = 25
Therefore, Julian is 25 years old.
2) Amanda has 20 coins in her purse. She has only 10c, 20c, and 50c coins, and their total value is $5. If she has more 50c than 10c coins, how many 10c coins does she have?
To approach this problem efficiently, we can use a combination of logic and basic arithmetic. Let's analyze the given information step by step:
We know that Amanda has a total of 20 coins.
Let's assume she has 'x' 10c coins, 'y' 20c coins, and 'z' 50c coins.
We are told that the total value of the coins is $5, so we can write the equation: 10x + 20y + 50z = 500 (converting dollars to cents).
Additionally, we know that Amanda has more 50c coins than 10c coins. So we can write the inequality: z > x.
Now, let's use the fact that Amanda has a total of 20 coins to establish another equation: x + y + z = 20.
By substituting the inequality z > x into the equation for a total of 20 coins, we get: x + y + z > x. Simplifying, we have: y + z > 0.
Since we need to find the number of 10c coins (x), we can rearrange the equation x + y + z = 20 to solve for x: x = 20 - y - z.
Now, let's simplify and substitute this expression for x into the equation for the total value: 10(20 - y - z) + 20y + 50z = 500.
Now, we can simplify and solve for y and z:
200 - 10y - 10z + 20y + 50z = 500
30y + 40z = 300
3y + 4z = 30
By plugging in different values for y and determining the corresponding values for z, we can find the solution that satisfies the given conditions (z > x and x + y + z = 20).
After evaluating the equation for various values of y, we find that when y = 2 and z = 6, the conditions are met:
x = 20 - y - z = 20 -2 -6 = 12
Therefore, Amanda has 12 coins of 10c.