Question

17. Admission prices to Cinema I to see a movie are $9.50 for an adult and $6.50 for a child. The admission charge at Cinema II is $8.00 per person regardless of age.

a. Write an inequality showing that the prices are cheaper at Cinema I than at Cinema II.
b. If 6 adults and their children go together to see a movie, use the inequality to find how many children must attend for Cinema I to be the better deal.

Answer 1

a. 9.5x + 6.5(x+c) < 8 when c>0

b. Must be one child more than the no. of adults.

Explanation:

For Cinema 1:

for adult = $9.50

for child = $6.50

For Cinema 2:

Per person regardless of age = $8.00

First of all, we will find out the condition when per person rates in both cinema are equal.

Assume x = no. of adults

y = no. of children

Rate per person in Cinema I = Rate per person in Cinema II

(9.5x + 6.5y)/(x+y) = 8

9.5x + 6.5y = 8(x+y)

9.5x + 6.5y = 8x + 8y

9.5x-8x = 8y-6.5y

=> x = y

So rates are equal when no. of adults equals no. of children

For Cinema I to have better rates, no. of children should be atleast 1 more than the no. of adult. In this way the rate per person of Cinema I will be less than 8

Hence we form an inequality when y = x+c and c > 0

9.5x + 6.5(x+c) < 8 when c>0

Hence there must be 1 more children than the no. of adults attending Cinema I for it to be a better deal.

Answer 2

9.50a+6.50c < 8.00(a+c), 6 children

Explanation:

Let the number of adults be a

the number of children be c

Part 1 :

Cinema 1:

Cost for adults = $9.50 x a = 9.50a

Cost of children = $6.50 x c = 6.50c

Total cost = 9.50a+6.50c

Cinema 2 :

Total cost = 8.00(a+c)

The inequality which shows that the cinema I is cheaper

9.50a+6.50c < 8.00(a+c)

9.50a+6.5c<8a+8c

9.5a-8a<8c-6.5c

1.5a<1.5c

a<c

Case 2:

6 adults goes to cinema , let they are accompanied by c number of children

Cinema 1

Total cost = 9.5 x 6 + 6.5 x c

for cinema 2 the total cost will be

8 ( 6+c)

for cinema 1 to be a better deal

9.5 x 6 + 6.5 x c < 8(6+c)

57+6.5c<48+8c

57-48<8c-6.5c

9<1.5c

c>6

Hence for Cinema 1 to be a better deal , there must be 6 children accompanying them