Agroup of people are waiting in line for a theater premiere. every 14th person in line will receive a free theater ticket. every 10th person will receive a gift card for $ 35. which person is the first to win both prizes? if there are 200 people in line, how many people will receive both prizes

a) 70th person

b) 2 people

Step-by-step explanation:

Requirement a)

To find the first person to get both a gift card and a free ticket, we have to use Least common multiples (L.C.M) of 10 and 14.

10 = 2 x 5

14 = 2 x 7

Therefore, the L.C.M of 10 and 14 is = 2 x 5 x 7 = 70

It means the first person will be the 70th person to win both a gift card and a free theater ticket.

Requirement b)

The required number of people to receive both the gifts will be the total number of people in line divided by the first person to achieve the feet.

Therefore,

When there are 70 people in line, both gifts to get by one person

When there are 1 people in line, both gifts to get by 1/70 person

When there are 200 people in line, both gifts to get by 200/70 person = 2.86 people (3 people).

However, since the number of people is 200, the next person to get the gifts will be 210. So, two people will receive both offerings.

option (b) is correct

step-by-step explanation:

to find the solution of this system, we need to convert it into reduced row-echelon form. we have,

after applying row operations r2 -> r2/2 and r3 -> r3/3, we get,