How many 20 Ω resistors must be connected in series to give an equivalent resistance to five 200 Ω resistors connected in parallel?
20x = 200/5
yes.
To find the number of 20 Ω resistors needed to give an equivalent resistance to five 200 Ω resistors connected in parallel, we can use the formula for resistors connected in parallel:
1/Req = 1/R1 + 1/R2 + 1/R3 + ...
Where Req is the equivalent resistance and R1, R2, R3, ... are the individual resistances.
Given that the individual resistances in parallel are 200 Ω and there are five of them, we can substitute these values into the formula:
1/Req = 1/200 + 1/200 + 1/200 + 1/200 + 1/200
1/Req = 5/200
Now, to find the equivalent resistance, we take the reciprocal of both sides of the equation:
Req = 200/5
Req = 40 Ω
Since the resistors in series add up, we need to find the number of 20 Ω resistors needed to give an equivalent resistance of 40 Ω:
Req = R1 + R2 + R3 + ...
40 = 20 + 20 + 20 + ...
Let x be the number of 20 Ω resistors needed:
40 = 20x
Dividing both sides by 20, we get:
x = 2
Therefore, the number of 20 Ω resistors that must be connected in series to give an equivalent resistance to the five 200 Ω resistors connected in parallel is 2.
To solve this problem, we first need to understand the formulas for resistors connected in series and parallel.
For resistors connected in series, the total resistance (R_total) is calculated by summing up the individual resistances:
R_total = R₁ + R₂ + R₃ + ...
For resistors connected in parallel, the total resistance (R_total) is calculated using the formula:
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...
Now let's solve the problem step by step:
1. Determine the equivalent resistance of the parallel combination:
Given that five 200 Ω resistors are connected in parallel, the formula for total resistance in parallel is:
1/R_total = 1/R₁ + 1/R₂ + 1/R₃ + ...
Substituting the given values:
1/R_total = 1/200 + 1/200 + 1/200 + 1/200 + 1/200
= 5/200
To calculate the equivalent resistance in parallel, take the reciprocal of both sides of the equation:
R_total = 200/5 = 40 Ω
2. Determine the number of resistors needed in series to achieve the equivalent resistance:
Since we want to achieve the same equivalent resistance using 20 Ω resistors connected in series, we can use the formula for total resistance in series:
R_total = R₁ + R₂ + R₃ + ...
Substituting the given value of equivalent resistance:
40 = 20n (where n is the number of 20 Ω resistors connected in series)
Solving for n, we divide both sides of the equation by 20:
n = 40/20 = 2
Therefore, you would need 2 twenty-ohm resistors connected in series to give an equivalent resistance to five 200 Ω resistors connected in parallel.