Question

1 find the lcd of each pair of fractions mentally.
a. \\(\frac{7}{8}, \frac{1}{2}\\) lcd =
b. \\(\frac{3}{5}, \frac{2}{3}\\) lcd =
c. \\(\frac{3}{4}, \frac{3}{10}\\) lcd =
d. \\(\frac{3}{7}, \frac{3}{14}, \frac{1}{2}\\) lcd =
e. \\(\frac{1}{3}, \frac{5}{6}, \frac{5}{8}\\) lcd =
f. \\(\frac{1}{2}, \frac{2}{3}, \frac{2}{5}\\) lcd =

2 add or subtract. simplify when possible.
a. \\(\

$$\begin{array}{r} \\frac{1}{2} \\\\ + \\frac{1}{3} \\\\ \\hline \\end{array}$$

\\)
b. \\(\

$$\begin{array}{r} \\frac{3}{4} \\\\ + \\frac{1}{8} \\\\ \\hline \\end{array}$$

\\)
c. \\(\

$$\begin{array}{r} \\frac{5}{6} \\\\ - \\frac{1}{4} \\\\ \\hline \\end{array}$$

\\)
d. \\(\begin{r} 3\frac{1}{2} \\\\ - 1\frac{1}{5} \\\\ \hline \end{array}\\)
e. \\(\

$$\begin{array}{r} \\frac{5}{6} \\\\ + \\frac{5}{9} \\\\ \\hline \\end{array}$$

\\)
f. \\(\

$$\begin{array}{r} 4\\frac{7}{9} \\\\ + 2\\frac{2}{3} \\\\ \\hline \\end{array}$$

\\)
g. \\(\

$$\begin{array}{r} 5\\frac{11}{15} \\\\ - 3\\frac{2}{5} \\\\ \\hline \\end{array}$$

\\)
h. \\(\

$$\begin{array}{r} 4\\frac{5}{8} \\\\ + 1\\frac{4}{7} \\\\ \\hline \\end{array}$$

\\)
i. \\(\

$$\begin{array}{r} 7\\frac{11}{12} \\\\ - 6\\frac{1}{4} \\\\ \\hline \\end{array}$$

\\)
j. \\(\

$$\begin{array}{r} 9\\frac{3}{4} \\\\ + 2\\frac{5}{9} \\\\ \\hline \\end{array}$$

\\)

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 16,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Least Common Denominator",
"Fraction Addition",
"Fraction Subtraction"
],
"current_concepts": [
"Least Common Denominator",
"Fraction Addition",
"Fraction Subtraction"
]
}
</pre_analysis>

<reasoning>

Find the LCD for each pair or group of fractions

\[

$$\begin{aligned} &\text{a. } \frac{7}{8}, \frac{1}{2} \implies \text{LCD} = \text{lcm}(8, 2) = 8\\ &\text{b. } \frac{3}{5}, \frac{2}{3} \implies \text{LCD} = \text{lcm}(5, 3) = 15\\ &\text{c. } \frac{3}{4}, \frac{3}{10} \implies \text{LCD} = \text{lcm}(4, 10) = 20\\ &\text{d. } \frac{3}{7}, \frac{3}{14}, \frac{1}{2} \implies \text{LCD} = \text{lcm}(7, 14, 2) = 14\\ &\text{e. } \frac{1}{3}, \frac{5}{6}, \frac{5}{8} \implies \text{LCD} = \text{lcm}(3, 6, 8) = 24\\ &\text{f. } \frac{1}{2}, \frac{2}{3}, \frac{2}{5} \implies \text{LCD} = \text{lcm}(2, 3, 5) = 30 \end{aligned}$$

\]

Add or subtract the fractions (a to e)

\[

$$\begin{aligned} &\text{a. } \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\\ &\text{b. } \frac{3}{4} + \frac{1}{8} = \frac{6}{8} + \frac{1}{8} = \frac{7}{8}\\ &\text{c. } \frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}\\ &\text{d. } 3\frac{1}{2} - 1\frac{1}{5} = 3\frac{5}{10} - 1\frac{2}{10} = 2\frac{3}{10}\\ &\text{e. } \frac{5}{6} + \frac{5}{9} = \frac{15}{18} + \frac{10}{18} = \frac{25}{18} = 1\frac{7}{18} \end{aligned}$$

\]

Add or subtract the fractions (f to j)

\[

$$\begin{aligned} &\text{f. } 4\frac{7}{9} + 2\frac{2}{3} = 4\frac{7}{9} + 2\frac{6}{9} = 6\frac{13}{9} = 7\frac{4}{9}\\ &\text{g. } 5\frac{11}{15} - 3\frac{2}{5} = 5\frac{11}{15} - 3\frac{6}{15} = 2\frac{5}{15} = 2\frac{1}{3}\\ &\text{h. } 4\frac{5}{8} + 1\frac{4}{7} = 4\frac{35}{56} + 1\frac{32}{56} = 5\frac{67}{56} = 6\frac{11}{56}\\ &\text{i. } 7\frac{11}{12} - 6\frac{1}{4} = 7\frac{11}{12} - 6\frac{3}{12} = 1\frac{8}{12} = 1\frac{2}{3}\\ &\text{j. } 9\frac{3}{4} + 2\frac{5}{9} = 9\frac{27}{36} + 2\frac{20}{36} = 11\frac{47}{36} = 12\frac{11}{36} \end{aligned}$$

\]
</reasoning>

<answer>

Section	Item	Answer
	b.	\(15\)
	c.	\(20\)
	d.	\(14\)
	e.	\(24\)
	f.	\(30\)
2. Add or subtract	a.	\(\frac{5}{6}\)
	b.	\(\frac{7}{8}\)
	c.	\(\frac{7}{12}\)
	d.	\(2\frac{3}{10}\)
	e.	\(1\frac{7}{18}\)
	f.	\(7\frac{4}{9}\)
	g.	\(2\frac{1}{3}\)
	h.	\(6\frac{11}{56}\)
	i.	\(1\frac{2}{3}\)
	j.	\(12\frac{11}{36}\)

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Arithmetic",
"Fraction Operations"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 16,
"skills_matched": [
"step_cot",
"table_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [],
"new_concepts": [
"Least Common Denominator",
"Fraction Addition",
"Fraction Subtraction"
],
"current_concepts": [
"Least Common Denominator",
"Fraction Addition",
"Fraction Subtraction"
]
}
</pre_analysis>

<reasoning>

Find the LCD for each pair or group of fractions

\[

$$\begin{aligned} &\text{a. } \frac{7}{8}, \frac{1}{2} \implies \text{LCD} = \text{lcm}(8, 2) = 8\\ &\text{b. } \frac{3}{5}, \frac{2}{3} \implies \text{LCD} = \text{lcm}(5, 3) = 15\\ &\text{c. } \frac{3}{4}, \frac{3}{10} \implies \text{LCD} = \text{lcm}(4, 10) = 20\\ &\text{d. } \frac{3}{7}, \frac{3}{14}, \frac{1}{2} \implies \text{LCD} = \text{lcm}(7, 14, 2) = 14\\ &\text{e. } \frac{1}{3}, \frac{5}{6}, \frac{5}{8} \implies \text{LCD} = \text{lcm}(3, 6, 8) = 24\\ &\text{f. } \frac{1}{2}, \frac{2}{3}, \frac{2}{5} \implies \text{LCD} = \text{lcm}(2, 3, 5) = 30 \end{aligned}$$

\]

Add or subtract the fractions (a to e)

\[

$$\begin{aligned} &\text{a. } \frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}\\ &\text{b. } \frac{3}{4} + \frac{1}{8} = \frac{6}{8} + \frac{1}{8} = \frac{7}{8}\\ &\text{c. } \frac{5}{6} - \frac{1}{4} = \frac{10}{12} - \frac{3}{12} = \frac{7}{12}\\ &\text{d. } 3\frac{1}{2} - 1\frac{1}{5} = 3\frac{5}{10} - 1\frac{2}{10} = 2\frac{3}{10}\\ &\text{e. } \frac{5}{6} + \frac{5}{9} = \frac{15}{18} + \frac{10}{18} = \frac{25}{18} = 1\frac{7}{18} \end{aligned}$$

\]

Add or subtract the fractions (f to j)

\[

$$\begin{aligned} &\text{f. } 4\frac{7}{9} + 2\frac{2}{3} = 4\frac{7}{9} + 2\frac{6}{9} = 6\frac{13}{9} = 7\frac{4}{9}\\ &\text{g. } 5\frac{11}{15} - 3\frac{2}{5} = 5\frac{11}{15} - 3\frac{6}{15} = 2\frac{5}{15} = 2\frac{1}{3}\\ &\text{h. } 4\frac{5}{8} + 1\frac{4}{7} = 4\frac{35}{56} + 1\frac{32}{56} = 5\frac{67}{56} = 6\frac{11}{56}\\ &\text{i. } 7\frac{11}{12} - 6\frac{1}{4} = 7\frac{11}{12} - 6\frac{3}{12} = 1\frac{8}{12} = 1\frac{2}{3}\\ &\text{j. } 9\frac{3}{4} + 2\frac{5}{9} = 9\frac{27}{36} + 2\frac{20}{36} = 11\frac{47}{36} = 12\frac{11}{36} \end{aligned}$$

\]
</reasoning>

<answer>

Section	Item	Answer
	b.	\(15\)
	c.	\(20\)
	d.	\(14\)
	e.	\(24\)
	f.	\(30\)
2. Add or subtract	a.	\(\frac{5}{6}\)
	b.	\(\frac{7}{8}\)
	c.	\(\frac{7}{12}\)
	d.	\(2\frac{3}{10}\)
	e.	\(1\frac{7}{18}\)
	f.	\(7\frac{4}{9}\)
	g.	\(2\frac{1}{3}\)
	h.	\(6\frac{11}{56}\)
	i.	\(1\frac{2}{3}\)
	j.	\(12\frac{11}{36}\)

</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Multi-part",
"knowledge_point": [
"Mathematics",
"Arithmetic",
"Fraction Operations"
]
}
</post_analysis>