Question 1 The sum of 4 two-digit and different positive even integers is 86. ¿What can be the largest of these numbers, yco? A) 30 B) 40 c) 50 D) 58 E) 64
Let's call the four numbers a, b, c, and d. Since the numbers are positive even integers, we can express them as follows:
a = 2x
b = 2y
c = 2z
d = 2w
where x, y, z, and w are positive integers.
The sum of these four numbers is 86:
a + b + c + d = 2x + 2y + 2z + 2w = 86
Dividing both sides by 2, we get:
x + y + z + w = 43
Since the numbers are two-digit, x, y, z, and w are all greater than or equal to 10. Therefore, the largest possible value for any of these numbers is 43.
The largest possible value for the largest number, yco, is equal to the value of d:
d = 2w = 2x = 2y = 2z = 43
Therefore, the largest possible value for yco is 43.
So, the correct answer is not provided in the options.
Let's work through this step-by-step:
Step 1: We are given that there are 4 two-digit positive even integers and their sum is 86.
Step 2: Let's assume the largest number is y.
Step 3: Since we have four two-digit positive even integers, we can assume the other three numbers are a, b, and c, where a < b < c < y.
Step 4: We can write the equation for the sum of the four numbers as y + c + b + a = 86.
Step 5: Since y is the largest number, it should be as large as possible. Therefore, we need to maximize y.
Step 6: The largest two-digit even number is 98.
Step 7: Substituting the values into the equation: 98 + c + b + a = 86.
Step 8: Rearranging the equation: c + b + a = 86 - 98 = -12.
Step 9: Since the numbers are positive, we cannot have a negative sum. Therefore, the largest number cannot be 98.
Step 10: Let's try the next largest even number, which is 96.
Step 11: Substituting the values into the equation: 96 + c + b + a = 86.
Step 12: Rearranging the equation: c + b + a = 86 - 96 = -10.
Step 13: Again, we cannot have a negative sum, so 96 cannot be the largest number.
Step 14: Let's try the next largest even number, which is 94.
Step 15: Substituting the values into the equation: 94 + c + b + a = 86.
Step 16: Rearranging the equation: c + b + a = 86 - 94 = -8.
Step 17: Once again, we cannot have a negative sum, so 94 cannot be the largest number.
Step 18: Let's try the next largest even number, which is 92.
Step 19: Substituting the values into the equation: 92 + c + b + a = 86.
Step 20: Rearranging the equation: c + b + a = 86 - 92 = -6.
Step 21: Once again, we cannot have a negative sum, so 92 cannot be the largest number.
Step 22: Finally, let's try the next largest even number, which is 90.
Step 23: Substituting the values into the equation: 90 + c + b + a = 86.
Step 24: Rearranging the equation: c + b + a = \86 - 90 = -4.
Step 25: Again, we cannot have a negative sum, so 90 cannot be the largest number.
Step 26: Based on our calculations, we cannot find a positive even number that satisfies the equation. Therefore, it is not possible to determine the largest number from the given options.
Answer: None of the options (A) 30, (B) 40, (C) 50, (D) 58, (E) 64 can be the largest number.
To find the largest possible number among the given options, we need to find the four two-digit even numbers whose sum is 86. Let's break down the problem step by step:
Step 1: Identify the even numbers
We are looking for even numbers, so we start with the smallest possibility, 2. We can represent even numbers as 2n, where n is an integer.
Step 2: Identify the two-digit numbers
The given problem specifies two-digit numbers. Since the numbers need to be even, the smallest two-digit even number is 20.
Step 3: Find the largest number
Now, we need to find four even two-digit numbers whose sum is 86. To maximize the largest number, we can start with the largest two-digit even number, which is 98 (2*49). Subtracting this from 86, we get 86 - 98 = -12, which is negative. This means we need to decrease the sum.
So, let's try with the next largest two-digit even number, 96 (2*48). Subtracting this from 86, we get 86 - 96 = -10, which is still negative.
Continuing this process, we find that subtracting 94 (2*47) from 86 gives us 86 - 94 = -8, which is still negative.
Finally, when we subtract 92 (2*46) from 86, we get 86 - 92 = -6, which is positive.
This means that the largest number is 92, and it can be represented as 2*46.
Therefore, the correct answer is option D) 58, as 92 is not among the given options.