Sender: "gayla werry"

My child is in the 6th grade and we are having difficulty with this question. Could you please help us. Thank you.

You have $1.00 to take 100 people to the movies. It costs: 1 cent for 10 children, 3 cents for each woman, 5 cents for each man. You have to take at least one of each category of people. Your answer has to add up to exactly 100 people and exactly $1.00. How many men, women, and children can you take?
Please explain your answer. Thank you.

Hi Gayla,

Suppose that you have the required collection of people. Put them into three categories, the men, the women and put the children in groups of 10. Suppose the number of men is m, the number of women is w and the number of groups of 10 children is g. There are two conditions to satisfy

  1. The total number of people is 100, that is

      10g + w + m = 100

  2. The cost is 100 cents, that is

      g + 3w + 5m = 100

Consider the men for a moment. There must be at least one man and there can't be 20 men or buying their tickets would take all the money. Thus the number of men is between 1 and 19. If you try some guesses you soon realize that the number of men must be large so lets try with m=19.

If m=19 then the 19 men pay 19*5=95 cents so, from the second condition, g + 3w = 5 cents. Hence w must be 1 and g must be 2. But g=2, w=3 and m=19 does not satisfy the first condition. Thus m is not 19.

If m=18 then the 18 men pay 18*5=90 cents. The two conditions are then

  1. The total number of people is 100, that is

      10g + w + 18 = 100

  2. The cost is 100 cents, that is

      g + 3w + 90 = 100

Since the units digit of 10g is zero the first condition implies that the units digit of w must be 2. If w=2 then the second condition implies that g=4, but g=4, w=2 and m=18 does not satisfy the first condition. Thus w=2 didn't work. You might try w=12 (remember the units digit must be 2) but the second condition shows that w=12 is too large. Thus m is not 18.

Not to give up lets try m=17. The 17 men then pay 5*17=85 cents and the two conditions become

  1. The total number of people is 100, that is

      10g + w + 17 = 100

  2. The cost is 100 cents, that is

      g + 3w + 85 = 100

This time the units digit of w is 3 but w=3 doesn't work and w=13 is too large.

Now try m=16.

Cheers,
Penny

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