215k views
5 votes
A singer has twice as many pants as pairs of shoes in her wardrobe and three-halves as many t-shirts as pants. What is the least number of clothing items she must have in her wardrobe in order to make different shoes, pants, and t-shirt combination each day of the year?

User Nemelis
by
8.3k points

1 Answer

6 votes

Answer:

24

Explanation:

Lets call p the total number of pants, s the number of pairs of shoes, and t the number of t-shirts. We have:

  • p = 2*s
  • t = 1.5*p = 1.5*(2*s) = 3*s

She can make each combination of clothing by selecting one pant, one pair of shoes and one t-shirt. Therefore, we know

  • She has as many possibilities of selecting pants as the amount of pants she has, thus she has 2*s possibilities.
  • She has s possibilities to pick a pair of shows
  • She has 3*s possibilities to pick a t-shirt.

As a result, she has a total of 2*s*s*3*s = s³*6 possibilities to choose. If she wants to make different combinations for an entire year, then 6s³ ≥ 366, as a result, s³ ≥ 61, so s ≥ ∛61 = 3.93

Since the number of clohes she has for each kind must be an integer, we know that s, 2*s and 3*s must be all integers, thus s must be at least 4.

As a verification, if she has 4 pair of shoes, then she must have 8 pants, and therefore 12 t-shirts, giving her a total of 4*8*12 = 384 possibilities. If she had just 3 pair of shoes, then she must had 6 pants and 9 t-shirts, and the total amount of possibilities might be only 3*6*9 = 162. Way below 365. We conclude that s = 4, p = 8, and t = 12.

The least number of clothes she must have should be 4+8+12 = 24

I hope this helped you!

User Prabhat Kasera
by
8.5k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.

9.4m questions

12.2m answers

Categories