Question

an amount of 100 dollars in dived into two first prizes of equal value and three second prices of equal value Each prize is a whole number of dollars and first prize is at least 4 times the value of the second prize if second prize is more that $6 find the amount of each prize

Answer 1

$38 and $8

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Let the first prize be f and the second prize be s.

According to the question we have the following conditions:

 1) 2f + 3s = 100;

 2) f and s are whole numbers;

 3) f ≥ 4s;

 4) s > 6.

First, from the equation 2f + 3s = 100 we can see s should be an even number, otherwise f can't be a whole number.

Next, let's assume s = 6, then:

• 2f + 3*6 = 100
• 2f + 18 = 100
• 2f = 82
• f = 41

It gives us that f should be less than 41.

Now, according to condition 3, we see that:

• f < 41 ⇒ 4s < 41 ⇒ s ≤ 10

Since s is even number,  s > 6 and s ≤ 10 it can be 8 or 10.

So, if s = 8 we find that:

• 2f + 3*8 = 100
• 2f = 76
• f = 38

If s = 10, we get:

• 2f + 3*10 = 100
• 2f = 70
• f = 35

Finally, let's verify the values against the conditions:

• f = 38, s = 8 - this pair meets all 4 conditions
• f = 35, s = 10 - all conditions are met apart from the third one since 35 < 4*10

It means the correct option is $38 and $8.

Answer 2

Final answer:

To divide $100 into two first prizes and three second prizes, with each first prize being at least four times the second prize (which is more than $6), we set up equations and inequalities based on the given conditions and iterate possible values to find whole number solutions for both prize amounts.

Explanation:

The question involves dividing an amount of $100 into two first prizes and three second prizes, where each prize amount is a whole number of dollars, and the first prize is at least four times greater than the second prize. The second prize also has to be more than $6. To find the amount of each prize, we can use algebra to set up equations that satisfy these conditions.

Let's denote the first prize as F and the second prize as S. The given conditions tell us that:

1. The total prize money is $100, which gives us the equation: 2F + 3S = $100.
2. The first prize is at least four times the second prize, which gives us the inequality: F ≥ 4S.
3. The second prize is more than $6, meaning S > 6.

In order to solve this, we can iterate over possible values for S (starting from $7 since it must be more than $6) and calculate the corresponding value of F using the total prize money equation, while checking if it satisfies the condition of F being at least four times greater than S.