SOLUTION: A cruise called ultimate comfort charges two different rates for seats. currently ˏ ratio of the number of tickets sold at the higher rate compared to the number of tickets

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Question 947879: A cruise called ultimate comfort charges two different rates for seats.
currently ˏ ratio of the number of tickets sold at the higher rate compared to the number of tickets sold at lower rate is 3:11.
however ˏonce the recession beganˏ2000 fewer tickets were sold at the higher rate and 1200 more tickets at the lower rate.
as a result the ratio during the recession was 1 : 5
find the number of each type of ticket sold originally

Found 2 solutions by Theo, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
let y = number of higher price tickets.
let x = number of lower price ticket.

during normal times the ratio of y/x is equal to 3/11.

solve for x in this equation to get x = (11*y)/3.

once the recession began, the ratio became:

(y-2000) / (x + 1200) = 1/5

solve for y in this equation to get y = 1/5 * (x + 1200) + 2000.

replace x with the value of x you solved for in the first equation to get:

y = 1/5 * ((11y)/3 + 1200) + 2000

multiply both sides of this equation by 5 to get:

5y = (11y)/3 + 1200 + 10000

multiply both side of this equation by 3 to get:

15y = 11y + 3600 + 30000

subtract 11y from both sides of this equation to get:

4y = 33600

divide both sides of this equation by 4 to get:

y = 8400

you have the value of y is equal to 8400

your original ratio was y/x = 3/11

replace y in this ratio to get 8400/x = 3/11

cross multiply to get 3x = 8400 * 11

divide both sides of this equation by 3 to get:

x = (8400 * 11) / 3

solve for x to get x = 30800.

in normal times y = 8400 and x = 30800.

the ratio of y to x is 8400 / 30800 which simplifies to 3/11.

since the recession, the ratio became (y-2000) / (x+1200) = 1/5

replace x with 30800 and y with 8400 in this ratio to get:

(8400 - 2000) / (30800 + 1200) = 1/5.

simplify to get 6400 / 32000 = 1/5.

simplify further to get 1/5 = 1/5.

this confirms that x = 8400 and y = 30800 is the correct number because all the ratios check out ok when using those numbers.

the solution to your problem is:

the number of each type of ticket sold originally is:

higher priced tickets = 8400
lower priced tickets = 30800.








Answer by MathTherapy(10556)   (Show Source): You can put this solution on YOUR website!
A cruise called ultimate comfort charges two different rates for seats.
currently ˏ ratio of the number of tickets sold at the higher rate compared to the number of tickets sold at lower rate is 3:11.
however ˏonce the recession beganˏ2000 fewer tickets were sold at the higher rate and 1200 more tickets at the lower rate.
as a result the ratio during the recession was 1 : 5
find the number of each type of ticket sold originally 
Let multiplicative factor for original 3:11 ratio, be x

Then, 3x of the higher rate tickets, and 11x of the lower rate tickets were sold, initially

When the recession began, 2,000 fewer higher-rate tickets were sold, so we get: 3x – 2,000

Also, 1,200 more lower-rate tickets were sold, so we get: 11x + 1,200

With recession ratio being  or , we get: 

5(3x – 2,000) = 1(11x + 1,200) ------- Cross-multiplying

15x – 10,000 = 11x + 1,200

15x – 11x = 1,200 + 10,000

4x = 11,200

x, or multiplicative factor = , or 2,800

Therefore, original number of higher-rate tickets sold were: 3(2,800), or  

Also, original number of lower-rate tickets sold were: 11(2,800), or  

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