Solve the ff. problems. Present a complete and systematic solution.
1.) A number added to twice its reciprocal is equal to 11/3. WHat is the number?
2.) The ratio of ten more than three times a number to the square of the same number is equal to 1. What is the number?
3.) The denominator of a fraction is 3 more than the numerator . If 5 is added to the numerator and 4 is subtracted from the denominator, the value of the new fraction is 2. What is the original fraction?
Use Quad. Formula to prob. 1, and 2.
1. x + 2*1/x = 11/3. x = ?.
2. (3x+10)/x^2 = 1. X = ?.
3. x+5)/(x+3-4) = 2. X = ?.
Original fraction = x/(x+3).
A number more than 32 devided by 15 more than twice a number is equal to 7/6
1.) To solve this problem, we need to set up an equation based on the given information. Let's call the number we're looking for "x".
According to the problem, "A number added to twice its reciprocal is equal to 11/3." In equation form, this can be written as:
x + 2(1/x) = 11/3
To solve this equation, we can follow these steps:
Step 1: Multiply the entire equation by 3x to eliminate the fractions:
3x(x) + 3x(2)(1/x) = 3x(11/3)
3x^2 + 6 = 11x
Step 2: Rearrange the equation into a quadratic equation:
3x^2 - 11x + 6 = 0
Step 3: Factor the quadratic equation or solve using the quadratic formula:
(x - 2)(3x - 3) = 0
This gives us two possible solutions: x = 2 or x = 1.
Therefore, the number that satisfies the equation is x = 2 or x = 1.
2.) Let's represent the number we're looking for as "x".
According to the problem, "The ratio of ten more than three times a number to the square of the same number is equal to 1." In equation form, this can be written as:
(3x + 10) / (x^2) = 1
To solve this equation, we can follow these steps:
Step 1: Multiply both sides of the equation by x^2 to eliminate the fraction:
(3x + 10) = x^2
Step 2: Rearrange the equation into a quadratic equation:
x^2 - 3x - 10 = 0
Step 3: Factor the quadratic equation or solve using the quadratic formula:
(x - 5)(x + 2) = 0
This gives us two possible solutions: x = 5 or x = -2.
However, we need to discard the negative value (-2) since the problem is asking for a number, and negative values usually do not make sense in this context.
Therefore, the number that satisfies the equation is x = 5.
3.) Let's represent the numerator as "x" and the denominator as "y".
According to the problem, "The denominator of a fraction is 3 more than the numerator." In equation form, this can be written as:
y = x + 3
And, "If 5 is added to the numerator and 4 is subtracted from the denominator, the value of the new fraction is 2." In equation form, this can be written as:
(x + 5) / (y - 4) = 2
To solve this system of equations, we can substitute the value of y from the first equation into the second equation:
(x + 5) / ((x + 3) - 4) = 2
(x + 5) / (x - 1) = 2
To eliminate the fraction, we can cross multiply:
2(x - 1) = x + 5
2x - 2 = x + 5
x = 7
Now, substitute the value of x back into the first equation to find y:
y = x + 3
y = 7 + 3
y = 10
Therefore, the original fraction is 7/10.