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4 of a number is added to 5/6. This is the same as 7/8 subtracted from the number. What is the number?

Mathematics

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Answer

416\frac{41}{6}

Step-by-step solution

7 steps

  1. Define the unknown number.

    Let the number be xx.

  2. Translate the first part of the statement into an algebraic expression.

    "34\frac{3}{4} of a number is added to 56\frac{5}{6}" can be written as:

    34x+56\frac{3}{4}x + \frac{5}{6}

  3. Translate the second part of the statement into an algebraic expression.

    "78\frac{7}{8} subtracted from the number" can be written as:

    x−78x - \frac{7}{8}

  4. Set up the equation based on the problem statement.

    The problem states that the results from Step 2 and Step 3 are the same:

    34x+56=x−78\frac{3}{4}x + \frac{5}{6} = x - \frac{7}{8}

  5. Solve the equation for xx.

    To eliminate the denominators, find the least common multiple (LCM) of 4, 6, and 8.
    The LCM of 4, 6, and 8 is 24.
    Multiply every term in the equation by 24:

    24(34x)+24(56)=24(x)−24(78)24 \left(\frac{3}{4}x\right) + 24 \left(\frac{5}{6}\right) = 24(x) - 24 \left(\frac{7}{8}\right) 6(3x)+4(5)=24x−3(7)6(3x) + 4(5) = 24x - 3(7) 18x+20=24x−2118x + 20 = 24x - 21

  6. Gather xx terms on one side and constant terms on the other side.

    Subtract 18x18x from both sides:

    20=24x−18x−2120 = 24x - 18x - 21 20=6x−2120 = 6x - 21

    Add 21 to both sides:

    20+21=6x20 + 21 = 6x 41=6x41 = 6x

  7. Isolate xx.

    Divide by 6:

    x=416x = \frac{41}{6}

    The number is 416\frac{41}{6}.

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