We are given three problems to solve. Problem 5: Find four consecutive numbers that sum to 22. Problem 6: Find three consecutive numbers such that three times the first, four times the second, and five times the third sums to 74. Problem 7: Find two consecutive even numbers such that five times the smaller exceeds three times the greater by 16.

1. Problem Description

We are given three problems to solve.

Problem 5: Find four consecutive numbers that sum to

2

2. Problem 6: Find three consecutive numbers such that three times the first, four times the second, and five times the third sums to

7

4. Problem 7: Find two consecutive even numbers such that five times the smaller exceeds three times the greater by

1

6.

2. Solution Steps

Problem 5:

Let the four consecutive numbers be $x, x+1, x+2, x+3$. Then their sum is $x + (x+1) + (x+2) + (x+3) = 22$.

Simplifying the equation, we get $4x + 6 = 22$.

Subtracting 6 from both sides, we have $4x = 16$.

Dividing both sides by 4, we get $x = 4$.

Therefore, the numbers are 4, 5, 6, and

7.

Problem 6:

Let the three consecutive numbers be $x, x+1, x+2$. Then, $3x + 4(x+1) + 5(x+2) = 74$.

Expanding the equation, we get $3x + 4x + 4 + 5x + 10 = 74$.

Combining like terms, we have $12x + 14 = 74$.

Subtracting 14 from both sides, we have $12x = 60$.

Dividing both sides by 12, we get $x = 5$.

Therefore, the numbers are 5, 6, and

7.

Problem 7:

Let the two consecutive even numbers be $2x$ and $2x+2$.

Then, $5(2x) = 3(2x+2) + 16$.

Simplifying the equation, we get $10x = 6x + 6 + 16$.

Combining like terms, we have $10x = 6x + 22$.

Subtracting $6x$ from both sides, we get $4x = 22$.

Dividing both sides by 4, we get $x = \frac{22}{4} = \frac{11}{2} = 5.5$.

Since $x$ must be an integer for $2x$ to be an even number, there is likely an error in the statement. Let the consecutive even integers be $x$ and $x+2$.

Then $5x = 3(x+2) + 16$

$5x = 3x + 6 + 16$

$5x = 3x + 22$

$2x = 22$

$x = 11$

$x+2 = 13$

However, these are not even numbers. Let us check our initial equations. Let us re-interpret the question: Find two consecutive even numbers such that five times the smaller number is 16 more than three times the greater number. Then, the equation is

$5x = 3(x+2) + 16$ as solved above, but x is meant to be even. So there are no even solutions. Let us try to interpret it as the other way. Find two consecutive even numbers such that five times the smaller exceeds three times the greater by

1

6. Thus $x, x+2$ are two consecutive even numbers.

$5x - 3(x+2) = 16$

$5x - 3x - 6 = 16$

$2x = 22$

$x = 11$

Again, this does not work. Let's assume there are integers such that $5x - 3(x+1) = 16$ which would mean consecutive integers (not even).

$5x - 3x - 3 = 16$

$2x = 19$ which means not integers exist.

Let's assume that the problem says 5 times the GREATER is 16 more than 3 times the smaller.

$5(2x+2) = 3(2x) + 16$

$10x + 10 = 6x + 16$

$4x = 6$

$x = \frac{3}{2}$

Which again means no integers can work.

The equation should be $5x = 3(x+2)+16$ where $x$ is even, but our calculations show no integers satisfy the equation, therefore there is likely a transcription error. Let's assume the problem intended consecutive integers $x$ and $x+1$ instead of consecutive even numbers. Then $5x=3(x+1)+16$ implies $5x=3x+3+16$ or $2x=19$, which does not lead to integers. Let us check if the problem was intending to read "...by 10." rather than "...by 16."

If this were true, we would have $5x = 3(x+2)+10$

$5x = 3x+6+10$

$2x = 16$

$x = 8$

Therefore $x+2=10$

Then if x=8 and x+2=10 are two consecutive even integers, $5*8=40$ and $3*10=30$ so $5x=3x+10$ is true. The consecutive even integers would be 8 and

1

0.

3. Final Answer

Problem 5: The numbers are 4, 5, 6, and

7. Problem 6: The numbers are 5, 6, and

7. Problem 7: Assuming the "...by 16" statement was in error and should be "...by 10", The numbers are 8 and

1

0. Otherwise, there are no such even numbers.