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Convert 101_two to base 10.

Chemistry

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Answer

C

Step-by-step solution

4 steps

To find the value of yy, we need to convert all the numbers in the equation to base 10. The equation is 101two+12y=23five101_{\text{two}} + 12_y = 23_{\text{five}}.

  1. Convert 101two101_{\text{two}} to base 10.

    101two=(1×22)+(0×21)+(1×20)101_{\text{two}} = (1 \times 2^2) + (0 \times 2^1) + (1 \times 2^0) =(1×4)+(0×2)+(1×1)= (1 \times 4) + (0 \times 2) + (1 \times 1) =4+0+1=5ten= 4 + 0 + 1 = 5_{\text{ten}}

  2. Convert 23five23_{\text{five}} to base 10.

    23five=(2×51)+(3×50)23_{\text{five}} = (2 \times 5^1) + (3 \times 5^0) =(2×5)+(3×1)= (2 \times 5) + (3 \times 1) =10+3=13ten= 10 + 3 = 13_{\text{ten}}

  3. Convert 12y12_y to base 10.

    12y=(1×y1)+(2×y0)12_y = (1 \times y^1) + (2 \times y^0) =(1×y)+(2×1)= (1 \times y) + (2 \times 1) =y+2ten= y + 2_{\text{ten}}

  4. Substitute the base 10 values into the original equation and solve for yy.

    5+(y+2)=135 + (y + 2) = 13 y+7=13y + 7 = 13

    Subtract 7 from both sides:

    y=13−7y = 13 - 7 y=6y = 6

    The value of yy is 6.
    Comparing this with the given options:
    A. 8
    B. 7
    C. 6
    D. 5

    The correct option is C.

    The final answer is C. 6\boxed{\text{C. } 6}.

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