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HOT Competition 28 Aug/8PM: When 120 candies are given to 25 children 28 Aug 2020, 20:00
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60% (02:09) correct 40% (02:27) wrong based on 589 sessions

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When 120 candies are given to 25 children and the number of candies for each child is at most 7, what is the maximum number of children who get only 1 candy?

A. 5
B. 6
C. 7
D. 8
E. 9


 

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E
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HOT Competition 28 Aug/8PM: When 120 candies are given to 25 children 28 Aug 2020, 20:29
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Given: When 120 candies are given to 25 children and the number of candies for each child is at most 7.
Asked: what is the maximum number of children who get only 1 candy?

Let us assume that x children receive only 1 candy. Remaining 25-x children will receive at most 7 candies each.

x+7(25-x)>=120
175-6x >=120
6x<= 55
x<=9.166
x_max =9

If 9 children each receive 1 candy only, 1 child receives 6 candies and 15 children receive 7 candies each, 120 candies will be consumed.

IMO E

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HOT Competition 28 Aug/8PM: When 120 candies are given to 25 children 28 Aug 2020, 20:17
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Quote:
When 120 candies are given to 25 children and the number of candies for each child is at most 7, what is the maximum number of children who get only 1 candy?

A. 5
B. 6
C. 7
D. 8
E. 9
Show more

The condition at which we can have maximum number of children getting 1 candy is when all the rest get the maximum possible (i.e. 7 in this case).
Let's now start analysing the answer choices.
We will start with the middle point C (this is because if it fails to satisfy, then we move up to B&A and if it does satisfy then we move down to C&D till we encounter a failure).
If 7 children get 1 candy, and I give 7 candy to all the others (i.e. 18), then we have a total of 7x1 + 18x7 = 133 candies. Way more than required. So check D next.
8 x 1 + 17 x 7 = 125 => still more than required. check E
9 x 1 + 16 x 7 = 121 => still satisfies. We have our answer at this stage (because this is a single option correct question and one of the answers has to be correct)

But just for sanity, check with 10 (we should fail):
10 x 1 + 15 x 7 = 115 (thus even by giving the maximum number of candies to all the other 15 children, we still fall short of the 120 candies mark. Hence, 9 is the maximum number that satisfied the condition).

Answer: E
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HOT Competition 28 Aug/8PM: When 120 candies are given to 25 children 29 Aug 2020, 02:27
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The answer is (E).

When 120 candies are given to 25 children and the number of candies for each child is at most 7, what is the maximum number of children who get only 1 candy?

Méthode 1


-> substitue choices to find max number. Since 9 is the biggest number, it would be better start from (E).

-> If each of 9 children get only 1 candy, 9 candies were distributed, and 111 candies and 16 children remained.
-> Can we distribute 111 candies to 16 children within the terms we were given? - yes.
-> We can give 7 candies to 15 children and 6 candies to remaining one children. (\(15\frac{6}{7}\))
-> Therefore 9 is the max number of children who get only 1 candy.

Methode 2


-> visualization

-> There is 25 children {??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??|??}
-> First, let’s distribute 1 candy to each child. {?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?|?}
-> We are left with 95 candies now.
-> Since we want to know max number of children who get 1 candy, let’s allow some kids to have max number of candies they can get.
Because if you distribute another round of 1 candy per kid equally, there wouldn’t be a kid with only one candy.
-> Since everyone has at least one candy in their hand, we can give out 6 candies per kid, and \(\frac{95}{6}=15\frac{5}{6}\)
{[?|?|?|?|?|?|?|?|?|?|?|?|?|?|?]|?|?|?|?|?|?|?|?|?|?}
-> Highlighted candies are children who got 7 candies in total, and underlined candy is a child who got 6 candies in total.
-> Therefore, 9 children left with only one candy in thier hand.


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HOT Competition 28 Aug/8PM: When 120 candies are given to 25 children 27 Sep 2021, 02:41
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Bunuel
When 120 candies are given to 25 children and the number of candies for each child is at most 7, what is the maximum number of children who get only 1 candy?

A. 5
B. 6
C. 7
D. 8
E. 9


 

This question was provided by Math Revolution
for the Heroes of Timers Competition

 

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Let us give 15 students 7 chocolates to maximise number of children recieving 1 chocolate
=>105

which leaves us with 15 chocolates to maximise weneed to give 1 child 6 toffes and remaining 9 children 1 toffe each

Therefore IMO E
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HOT Competition 28 Aug/8PM: When 120 candies are given to 25 children 09 Sep 2025, 10:54
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Bunuel
When 120 candies are given to 25 children and the number of candies for each child is at most 7, what is the maximum number of children who get only 1 candy?

A. 5
B. 6
C. 7
D. 8
E. 9


 

This question was provided by Math Revolution
for the Heroes of Timers Competition

 

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Give 1 candy to each person.
95 candies will be left.
No since each person already has 1 candy the maximum we can assign anyone is 7 candies, we will have to give 6 more to maximum people.
95/6 = 15
Now, 90 candies are gone, and 5 are left, we can give this to some other person
Therefore, 15 people have 7 candies, 1 person has 6 candies and the rest have 1 candy.
25 - (15+1)
= 9 people
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