At a concert, tickets were sold at $12 , $30 and $50. The sales of $12 tickets made up 25% of the total amout of money collected. The sales of $30 tickets made up of 60% of the total amount . Given that 5 more $12 tickets were sold than the $30 tickets , how many $50 tickets were sold?
The ans is 18 but i don't understand the method in my book
neither do I. But, if the amounts of the tickets are x,y,z, then
12x = (12x+30y+50z)(1/4)
30y = (12x+30y+50z)(3/5)
x = y+5
Clear the fractions and do some substitution, and you will find that z=18
To solve this problem, we can set up a system of equations based on the given information.
Let's start by assigning variables to the unknown quantities:
Let x be the number of $12 tickets sold.
Let y be the number of $30 tickets sold.
Let z be the number of $50 tickets sold.
Given that the sales of $12 tickets made up 25% of the total amount, we can write the equation:
12x = 0.25(Total Amount) -- Equation 1
Similarly, the sales of $30 tickets made up 60% of the total amount, so we can write:
30y = 0.60(Total Amount) -- Equation 2
We also know that 5 more $12 tickets were sold than the $30 tickets. This gives us the equation:
x = y + 5 -- Equation 3
Now, let's solve this system of equations step by step:
Step 1: Substitute equation 3 into equation 1.
12(y + 5) = 0.25(Total Amount)
12y + 60 = 0.25(Total Amount)
Step 2: Substitute equation 3 into equation 2.
30y = 0.60(Total Amount)
Step 3: Set the left sides of equations 2 and 3 equal to each other since they both represent 0.60(Total Amount).
12y + 60 = 30y
Step 4: Subtract 12y from both sides of the equation.
60 = 18y
Step 5: Divide both sides by 18 to solve for y.
y = 60 / 18
y = 3.33 (approximately)
Since we can't have a fraction of a ticket, we'll round down y to the nearest whole number:
y = 3
Step 6: Substitute the value of y back into equation 3 to solve for x.
x = y + 5
x = 3 + 5
x = 8
So, 8 $12 tickets were sold.
Step 7: Substitute the values of x and y into equation 2 to solve for the total amount.
30y = 0.60(Total Amount)
30 * 3 = 0.60(Total Amount)
90 = 0.60(Total Amount)
Step 8: Divide both sides by 0.60 to solve for the total amount.
Total Amount = 90 / 0.60
Total Amount = 150
Step 9: Substitute the values of x, y, and the total amount into equation 1 to solve for z.
12x = 0.25(Total Amount)
12 * 8 = 0.25(150)
96 = 0.25(150)
Step 10: Divide both sides by 0.25 to solve for z.
z = 96 / 0.25
z = 384
So, 384 $50 tickets were sold.
Therefore, the number of $50 tickets sold is 384.