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Simplify ((1)/(2) + 2(1)/(2)) - (5(1)/(2) - 1(1)/(2)) (b) Solve for n in the equation 2n2 = 4n. (c) Use logarithm tables to evaluate (734.6)/(56.4).

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3 parts
a

−12\mathbf{-\frac{1}{2}}

b

n=0 or n=2\mathbf{n=0 \text{ or } n=2}

c

13.02\mathbf{13.02}

Step-by-step solution

12 steps

Here are the solutions to the problems.

Part a· 5 steps

Simplify: (112+214)−(512−114)(1\frac{1}{2} + 2\frac{1}{4}) - (5\frac{1}{2} - 1\frac{1}{4})

  1. Convert all mixed numbers to improper fractions.

    112=321\frac{1}{2} = \frac{3}{2} 214=942\frac{1}{4} = \frac{9}{4} 512=1125\frac{1}{2} = \frac{11}{2} 114=541\frac{1}{4} = \frac{5}{4}

  2. Perform the addition inside the first parenthesis.

    (32+94)=(64+94)=154\left(\frac{3}{2} + \frac{9}{4}\right) = \left(\frac{6}{4} + \frac{9}{4}\right) = \frac{15}{4}

  3. Perform the subtraction inside the second parenthesis.

    (112−54)=(224−54)=174\left(\frac{11}{2} - \frac{5}{4}\right) = \left(\frac{22}{4} - \frac{5}{4}\right) = \frac{17}{4}

  4. Perform the final subtraction.

    154−174=−24\frac{15}{4} - \frac{17}{4} = -\frac{2}{4}

  5. Simplify the fraction.

    −24=−12-\frac{2}{4} = -\frac{1}{2} −12\boxed{\mathbf{-\frac{1}{2}}}

Part b

n=0 or n=2\mathbf{n=0 \text{ or } n=2}

Solve for nn in the equation 2n2=4n2n^2 = 4n.

  1. Move all terms to one side to set the equation to zero.

    2n2−4n=02n^2 - 4n = 0

  2. Factor out the common term, $2n$.

    2n(n−2)=02n(n - 2) = 0

  3. Set each factor equal to zero and solve for nn.

    2n=0  ⟹  n=02n = 0 \implies n = 0 n−2=0  ⟹  n=2n - 2 = 0 \implies n = 2 n=0 or n=2\boxed{\mathbf{n=0 \text{ or } n=2}}

Part c

13.02\mathbf{13.02}

Use logarithm tables to evaluate: 734.656.4\frac{734.6}{56.4}

Let X=734.656.4X = \frac{734.6}{56.4}. We will use the property log⁡(ab)=log⁡a−log⁡b\log\left(\frac{a}{b}\right) = \log a - \log b.

  1. Find the logarithm of 734.6.

    The characteristic is 2 (since there are 3 digits before the decimal point, 3−1=23-1=2).
    Using logarithm tables, the mantissa for 734.6 is approximately 0.8660.
    So, log⁡(734.6)=2.8660\log(734.6) = 2.8660.

  2. Find the logarithm of 56.4.

    The characteristic is 1 (since there are 2 digits before the decimal point, 2−1=12-1=1).
    Using logarithm tables, the mantissa for 56.4 is approximately 0.7513.
    So, log⁡(56.4)=1.7513\log(56.4) = 1.7513.

  3. Subtract the logarithms.

    log⁡X=log⁡(734.6)−log⁡(56.4)\log X = \log(734.6) - \log(56.4) log⁡X=2.8660−1.7513\log X = 2.8660 - 1.7513 log⁡X=1.1147\log X = 1.1147

  4. Find the antilogarithm of 1.1147.

    The characteristic is 1, which means the number has 1+1=21+1=2 digits before the decimal point.
    Using antilogarithm tables, the antilog of the mantissa 0.1147 is approximately 1.302.
    Therefore, X=13.02X = 13.02.

    13.02\boxed{\mathbf{13.02}}
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