We are presented with three separate math problems. Let's define $x$ as the unknown number for each problem. Problem 2: If we multiply a number by 6 and add 1, we get the same answer as when we add 5 to the number and then treble the result. Find this number. Problem 3: If we add 3 to a number and quadruple the result, the answer is 36. What is the number? Problem 4: If you add 5 to a number and then multiply the result by 4, you get the same answer as when you add 1 to the number and then multiply the result by 2. What is the number?

1. Problem Description

We are presented with three separate math problems. Let's define $x$ as the unknown number for each problem.

Problem 2: If we multiply a number by 6 and add 1, we get the same answer as when we add 5 to the number and then treble the result. Find this number.

Problem 3: If we add 3 to a number and quadruple the result, the answer is

3

6. What is the number?

Problem 4: If you add 5 to a number and then multiply the result by 4, you get the same answer as when you add 1 to the number and then multiply the result by

2. What is the number?

2. Solution Steps

Problem 2:

* Translate the word problem into an equation: $6x + 1 = 3(x + 5)$

* Expand the right side: $6x + 1 = 3x + 15$

* Subtract $3x$ from both sides: $3x + 1 = 15$

* Subtract 1 from both sides: $3x = 14$

* Divide both sides by 3: $x = \frac{14}{3}$

Problem 3:

* Translate the word problem into an equation: $4(x + 3) = 36$

* Expand the left side: $4x + 12 = 36$

* Subtract 12 from both sides: $4x = 24$

* Divide both sides by 4: $x = 6$

Problem 4:

* Translate the word problem into an equation: $4(x + 5) = 2(x + 1)$

* Expand both sides: $4x + 20 = 2x + 2$

* Subtract $2x$ from both sides: $2x + 20 = 2$

* Subtract 20 from both sides: $2x = -18$

* Divide both sides by 2: $x = -9$

3. Final Answer

Problem 2: $\frac{14}{3}$

Problem 3: $6$

Problem 4: $-9$