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s = 2b = 5t = 7n = 1

3t+2

NoCalc
s+2b

NoCalc
5b+n

NoCalc
7t−2s

NoCalc
tb

NoCalc
tn−1

NoCalc
20s−b

NoCalc
tn+sb

NoCalc
5t−b−2s

NoCalc
nbt−s

12s−2bt+101

9n+8b+7s−6t

1 answer

Given the equations provided:

  1. \( s = 2b = 5t = 7n = 1 \)

Let's solve for \( s, b, t, n \).

From \( s = 1 \):

  • Since \( s = 2b \), we have \( 2b = 1 \) → \( b = \frac{1}{2} \).
  • Since \( s = 5t \), we have \( 5t = 1 \) → \( t = \frac{1}{5} \).
  • Since \( s = 7n \), we have \( 7n = 1 \) → \( n = \frac{1}{7} \).

Now we can plug these values into each expression:

  1. For \( 3t + 2 \): \[ 3t + 2 = 3\left(\frac{1}{5}\right) + 2 = \frac{3}{5} + 2 = \frac{3}{5} + \frac{10}{5} = \frac{13}{5} \]

  2. For \( s + 2b \): \[ s + 2b = 1 + 2\left(\frac{1}{2}\right) = 1 + 1 = 2 \]

  3. For \( 5b + n \): \[ 5b + n = 5\left(\frac{1}{2}\right) + \frac{1}{7} = \frac{5}{2} + \frac{1}{7} \] To add these, find a common denominator (14): \[ = \frac{35}{14} + \frac{2}{14} = \frac{37}{14} \]

  4. For \( 7t - 2s \): \[ 7t - 2s = 7\left(\frac{1}{5}\right) - 2(1) = \frac{7}{5} - 2 = \frac{7}{5} - \frac{10}{5} = -\frac{3}{5} \]

  5. For \( tb \): \[ tb = t \cdot b = \left(\frac{1}{5}\right)\left(\frac{1}{2}\right) = \frac{1}{10} \]

  6. For \( tn - 1 \): \[ tn - 1 = t \cdot n - 1 = \left(\frac{1}{5}\right)\left(\frac{1}{7}\right) - 1 = \frac{1}{35} - 1 = \frac{1}{35} - \frac{35}{35} = -\frac{34}{35} \]

  7. For \( 20s - b \): \[ 20s - b = 20(1) - \frac{1}{2} = 20 - \frac{1}{2} = 20 - 0.5 = 19.5 \]

  8. For \( tn + sb \): \[ tn + sb = t \cdot n + s \cdot b = \left(\frac{1}{5}\right)\left(\frac{1}{7}\right) + (1)\left(\frac{1}{2}\right) = \frac{1}{35} + \frac{1}{2} \] Common denominator (70): \[ = \frac{2}{70} + \frac{35}{70} = \frac{37}{70} \]

  9. For \( 5t - b - 2s \): \[ 5t - b - 2s = 5\left(\frac{1}{5}\right) - \frac{1}{2} - 2(1) = 1 - \frac{1}{2} - 2 = 1 - 0.5 - 2 = -1.5 = -\frac{3}{2} \]

  10. For \( nbt - s \): \[ nbt - s = n \cdot b \cdot t - s = \left(\frac{1}{7}\right)\left(\frac{1}{2}\right)\left(\frac{1}{5}\right) - 1 = \frac{1}{70} - 1 = \frac{1}{70} - \frac{70}{70} = -\frac{69}{70} \]

  11. For \( 12s - 2bt + 101 \): \[ 12s - 2bt + 101 = 12(1) - 2\left(\frac{1}{2}\right)\left(\frac{1}{5}\right) + 101 = 12 - \frac{1}{5} + 101 \] \[ = 112 - \frac{1}{5} = 112 - 0.2 = 111.8 = \frac{559}{5} \]

  12. For \( 9n + 8b + 7s - 6t \): \[ 9n + 8b + 7s - 6t = 9\left(\frac{1}{7}\right) + 8\left(\frac{1}{2}\right) + 7(1) - 6\left(\frac{1}{5}\right) \] \[ = \frac{9}{7} + 4 + 7 - \frac{6}{5} \] To combine these, convert each to a common denominator (35): \[ = \frac{45}{35} + \frac{140}{35} + \frac{245}{35} - \frac{42}{35} = \frac{45 + 140 + 245 - 42}{35} = \frac{388}{35} \]

Here are all the final answers summarized:

  1. \( 3t + 2 = \frac{13}{5} \)
  2. \( s + 2b = 2 \)
  3. \( 5b + n = \frac{37}{14} \)
  4. \( 7t - 2s = -\frac{3}{5} \)
  5. \( tb = \frac{1}{10} \)
  6. \( tn - 1 = -\frac{34}{35} \)
  7. \( 20s - b = 19.5 \)
  8. \( tn + sb = \frac{37}{70} \)
  9. \( 5t - b - 2s = -\frac{3}{2} \)
  10. \( nbt - s = -\frac{69}{70} \)
  11. \( 12s - 2bt + 101 = \frac{559}{5} \)
  12. \( 9n + 8b + 7s - 6t = \frac{388}{35} \)