Question

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

Answer 1

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. \( 9n + 8b + 7s - 6t = \frac{388}{35} \)