Question: Which expressions are equivalent to One-third + 8 d + StartFraction 5 Over 7 EndFraction? Check all that apply.
1 + 3 + 8 d + 5 + 7
8 d + one-third + StartFraction 5 Over 7 EndFraction
StartFraction 1 Over 7 EndFraction + 8 d + StartFraction 5 Over 3 EndFraction
One-third + StartFraction 5 Over 7 EndFraction + 8 d
3 + 8 d + 7
8 d + one-third + StartFraction 7 Over 5 EndFraction
StartFraction 5 Over 7 EndFraction + one-third + 8 d
1. **State the problem:** We want to find which expressions are equivalent to $$\frac{1}{3} + 8d + \frac{5}{7}$$.
2. **Recall the properties:** Addition is commutative and associative, so terms can be rearranged without changing the value.
3. **Analyze each option:**
- Option 1: $$1 + 3 + 8d + 5 + 7$$
- Sum constants: $$1 + 3 + 5 + 7 = 16$$
- Expression becomes $$16 + 8d$$, which is not equivalent to $$\frac{1}{3} + 8d + \frac{5}{7}$$.
- Option 2: $$8d + \text{one-third} + \frac{5}{7}$$
- This is exactly $$8d + \frac{1}{3} + \frac{5}{7}$$, which is equivalent by commutativity.
- Option 3: $$\frac{1}{7} + 8d + \frac{5}{3}$$
- The fractions $$\frac{1}{7}$$ and $$\frac{5}{3}$$ differ from $$\frac{1}{3}$$ and $$\frac{5}{7}$$, so not equivalent.
- Option 4: $$\text{one-third} + \frac{5}{7} + 8d$$
- Same terms as original, just reordered, so equivalent.
- Option 5: $$3 + 8d + 7$$
- Sum constants: $$3 + 7 = 10$$
- Expression is $$10 + 8d$$, not equivalent.
- Option 6: $$8d + \text{one-third} + \frac{7}{5}$$
- Fraction $$\frac{7}{5}$$ differs from $$\frac{5}{7}$$, so not equivalent.
- Option 7: $$\frac{5}{7} + \text{one-third} + 8d$$
- Same terms as original, reordered, so equivalent.
4. **Final answer:** The equivalent expressions are options 2, 4, and 7.
**Summary:**
$$\frac{1}{3} + 8d + \frac{5}{7} = 8d + \frac{1}{3} + \frac{5}{7} = \frac{1}{3} + \frac{5}{7} + 8d = \frac{5}{7} + \frac{1}{3} + 8d$$