Learn how to solve word problems involving fractions and equations, and how to evaluate an infinite continued fraction. This video provides step-by-step solutions and explanations. Perfect for students learning algebra! #algebra #wordproblems #continuedfractions #equations #math #tutorial

𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Find the number which when added to its two thirds is equal to 35.
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
Let the number be x.
According to the problem statement, the number added to its two thirds is equal to 35. We can write this as an equation:
x + (2/3)x = 35
To solve for x, first find a common denominator for the terms on the left side:
(3/3)x + (2/3)x = 35
(3x + 2x) / 3 = 35
5x / 3 = 35
Now, multiply both sides of the equation by 3 to isolate the term with x:
5x = 35 * 3
5x = 105
Finally, divide both sides by 5 to solve for x:
x = 105 / 5
x = 21
The number is 21.
Final Answer: 21

𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: Solve (12x+1)/4 = (13x-1)/5 + 3
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
To solve this equation, first eliminate the fractions by multiplying both sides by the least common multiple (LCM) of the denominators, which is LCM(4, 5) = 20.
20 * [(12x+1)/4] = 20 * [(13x-1)/5 + 3]
5 * (12x + 1) = 4 * (13x - 1) + 20 * 3
60x + 5 = 52x - 4 + 60
60x + 5 = 52x + 56
Now, rearrange the equation to group the x terms on one side and the constant terms on the other side:
60x - 52x = 56 - 5
8x = 51
Finally, divide by 8 to solve for x:
x = 51 / 8
Final Answer: x = 51/8

𝐐𝐮𝐞𝐬𝐭𝐢𝐨𝐧: What is the value of 4 + 1/(4 + 1/(4 + 1/(4 + ...∞) ) )
𝐒𝐨𝐥𝐮𝐭𝐢𝐨𝐧:
Let the given expression be equal to y:
y = 4 + 1/(4 + 1/(4 + 1/(4 + ...∞) ) )
Notice that the part after the initial '4' is the same as the original expression y. So, we can rewrite the equation as:
y = 4 + 1/y
Now, we need to solve this equation for y. Multiply both sides by y:
y * y = 4 * y + (1/y) * y
y² = 4y + 1
Rearrange the equation into a standard quadratic form:
y² - 4y - 1 = 0
We can solve this quadratic equation using the quadratic formula:
y = [-b ± √(b² - 4ac)] / 2a
Here, a = 1, b = -4, and c = -1.
y = [-(-4) ± √((-4)² - 4(1)(-1))] / 2(1)
y = [4 ± √(16 + 4)] / 2
y = [4 ± √20] / 2
y = [4 ± √(4 * 5)] / 2
y = [4 ± 2√5] / 2
y = 2 ± √5
Since the expression is a sum of positive numbers, its value must be positive.
√5 is approximately 2.236.
So, 2 + √5 ≈ 2 + 2.236 = 4.236 (positive)
And, 2 - √5 ≈ 2 - 2.236 = -0.236 (negative)
Therefore, the value of the expression is 2 + √5.
Final Answer: 2 + √5

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