Here are the step-by-step solutions for each equation on the worksheet.
1) $\frac{1}{8} = \frac{k - 10}{k}$
* Cross-multiply: $1 \cdot k = 8 \cdot (k - 10)$
* Simplify: $k = 8k - 80$
* Subtract $8k$ from both sides: $-7k = -80$
* Divide by $-7$: $k = \frac{80}{7}$
* Calculate decimal: $k \approx 11.428...$
* Round to nearest hundredth:
11.43
2) $\frac{3}{c} = \frac{8}{28}$
* Cross-multiply: $3 \cdot 28 = 8 \cdot c$
* Simplify: $84 = 8c$
* Divide by 8: $c = \frac{84}{8}$
* Calculate decimal: $c = 10.5$
* Answer:
10.5
3) $\frac{3}{d + 11} = \frac{7}{d - 24}$
* Cross-multiply: $3(d - 24) = 7(d + 11)$
* Distribute: $3d - 72 = 7d + 77$
* Subtract $3d$ from both sides: $-72 = 4d + 77$
* Subtract 77 from both sides: $-149 = 4d$
* Divide by 4: $d = -\frac{149}{4}$
* Calculate decimal: $d = -37.25$
* Answer:
-37.25
4) $\frac{y + 10}{14} = \frac{4}{5}$
* Cross-multiply: $5(y + 10) = 4 \cdot 14$
* Simplify right side: $5(y + 10) = 56$
* Distribute left side: $5y + 50 = 56$
* Subtract 50 from both sides: $5y = 6$
* Divide by 5: $y = \frac{6}{5}$
* Calculate decimal: $y = 1.2$
* Answer:
1.2
5) $\frac{1}{8} = \frac{b + 27}{b - 23}$
* Cross-multiply: $1(b - 23) = 8(b + 27)$
* Distribute: $b - 23 = 8b + 216$
* Subtract $b$ from both sides: $-23 = 7b + 216$
* Subtract 216 from both sides: $-239 = 7b$
* Divide by 7: $b = -\frac{239}{7}$
* Calculate decimal: $b \approx -34.142...$
* Round to nearest hundredth:
-34.14
6) $\frac{4}{9} = \frac{26}{f + 7}$
* Cross-multiply: $4(f + 7) = 9 \cdot 26$
* Simplify right side: $4(f + 7) = 234$
* Distribute left side: $4f + 28 = 234$
* Subtract 28 from both sides: $4f = 206$
* Divide by 4: $f = \frac{206}{4}$
* Calculate decimal: $f = 51.5$
* Answer:
51.5
7) $\frac{1}{9} = \frac{n}{n - 15}$
* Cross-multiply: $1(n - 15) = 9n$
* Simplify: $n - 15 = 9n$
* Subtract $n$ from both sides: $-15 = 8n$
* Divide by 8: $n = -\frac{15}{8}$
* Calculate decimal: $n = -1.875$
* Round to nearest hundredth:
-1.88
8) $\frac{6}{7} = \frac{z}{z - 28}$
* Cross-multiply: $6(z - 28) = 7z$
* Distribute: $6z - 168 = 7z$
* Subtract $6z$ from both sides: $-168 = z$
* Answer:
-168
9) $\frac{a + 21}{7} = \frac{a - 3}{8}$
* Cross-multiply: $8(a + 21) = 7(a - 3)$
* Distribute: $8a + 168 = 7a - 21$
* Subtract $7a$ from both sides: $a + 168 = -21$
* Subtract 168 from both sides: $a = -189$
* Answer:
-189
10) $\frac{19}{x + 8} = \frac{1}{8}$
* Cross-multiply: $19 \cdot 8 = 1(x + 8)$
* Simplify: $152 = x + 8$
* Subtract 8 from both sides: $144 = x$
* Answer:
144
Final Answer:
1) 11.43
2) 10.5
3) -37.25
4) 1.2
5) -34.14
6) 51.5
7) -1.88
8) -168
9) -189
10) 144
Parent Tip: Review the logic above to help your child master the concept of basic algebra worksheet with answer keys.
