Set of 14 algebra and fraction simplification problems with solutions, ideal for math practice.

It looks like you've uploaded an image containing a list of 14 mathematical problems, mostly involving fraction arithmetic and algebraic simplification. Each problem has a solution written in red next to it. I'll go through each one step by step, verify the solutions, and explain how to solve them.

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🔢 Problem 1:

$$
\frac{2}{3} + \frac{3}{4} = \frac{85}{12}
$$

Check:
Find common denominator: LCM(3,4) = 12
$$
\frac{2}{3} = \frac{8}{12},\quad \frac{3}{4} = \frac{9}{12}
\Rightarrow \frac{8}{12} + \frac{9}{12} = \frac{17}{12}
$$

But the given answer is $ \frac{85}{12} $ — this is incorrect.

✔ Correct Answer: $ \frac{17}{12} $

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🔢 Problem 2:

$$
\frac{3}{4} - \frac{1}{3} = \frac{5}{8}
$$

LCM(4,3) = 12
$$
\frac{3}{4} = \frac{9}{12},\quad \frac{1}{3} = \frac{4}{12}
\Rightarrow \frac{9}{12} - \frac{4}{12} = \frac{5}{12}
$$

Given answer: $ \frac{5}{8} $ → Incorrect

✔ Correct Answer: $ \frac{5}{12} $

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🔢 Problem 3:

$$
\frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{65}{132}
$$

LCM(2,3,4) = 12
$$
\frac{1}{2} = \frac{6}{12},\quad \frac{1}{3} = \frac{4}{12},\quad \frac{1}{4} = \frac{3}{12}
\Rightarrow \frac{6+4+3}{12} = \frac{13}{12}
$$

Given: $ \frac{65}{132} $ → Simplify: $ \frac{65}{132} \approx 0.492 $, but $ \frac{13}{12} \approx 1.083 $ → Wrong

✔ Correct Answer: $ \frac{13}{12} $

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🔢 Problem 4:

$$
\frac{4}{5} \div \left( \frac{1}{5} + \frac{2}{3} \right) = \frac{12}{13}
$$

First simplify inside parentheses:
$$
\frac{1}{5} + \frac{2}{3} = \frac{3}{15} + \frac{10}{15} = \frac{13}{15}
$$

Now divide:
$$
\frac{4}{5} \div \frac{13}{15} = \frac{4}{5} \times \frac{15}{13} = \frac{60}{65} = \frac{12}{13}
$$

✔ Correct! Answer: $ \frac{12}{13} $

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🔢 Problem 5:

$$
\frac{5 - \frac{2}{5}}{6 + \frac{1}{3}} = \frac{69}{95}
$$

Numerator: $ 5 - \frac{2}{5} = \frac{25}{5} - \frac{2}{5} = \frac{23}{5} $

Denominator: $ 6 + \frac{1}{3} = \frac{18}{3} + \frac{1}{3} = \frac{19}{3} $

Now divide:
$$
\frac{23/5}{19/3} = \frac{23}{5} \times \frac{3}{19} = \frac{69}{95}
$$

✔ Correct! Answer: $ \frac{69}{95} $

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🔢 Problem 6:

$$
\frac{1}{2} + \frac{3}{5} + \frac{1}{3} = \frac{101}{92}
$$

Wait — this is not correct.

LCM(2,5,3) = 30
$$
\frac{1}{2} = \frac{15}{30},\quad \frac{3}{5} = \frac{18}{30},\quad \frac{1}{3} = \frac{10}{30}
\Rightarrow \frac{15+18+10}{30} = \frac{43}{30}
$$

Given: $ \frac{101}{92} \approx 1.1 $, but $ \frac{43}{30} \approx 1.433 $ → Incorrect

✔ Correct Answer: $ \frac{43}{30} $

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🔢 Problem 7:

$$
\frac{x + \frac{y}{4}}{x - \frac{y}{3}} = \frac{3(4x + y)}{4(3x - y)}
$$

Let’s simplify numerator and denominator:

Numerator: $ x + \frac{y}{4} = \frac{4x + y}{4} $

Denominator: $ x - \frac{y}{3} = \frac{3x - y}{3} $

So:
$$
\frac{\frac{4x + y}{4}}{\frac{3x - y}{3}} = \frac{4x + y}{4} \cdot \frac{3}{3x - y} = \frac{3(4x + y)}{4(3x - y)}
$$

✔ Correct! Answer: $ \frac{3(4x + y)}{4(3x - y)} $

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🔢 Problem 8:

$$
\frac{\frac{a}{b} + \frac{x}{y}}{\frac{a}{z} - \frac{x}{c}} = \frac{cz(ay + bx)}{by(ac - xz)}
$$

We simplify numerator and denominator separately.

Numerator:
$$
\frac{a}{b} + \frac{x}{y} = \frac{ay + bx}{by}
$$

Denominator:
$$
\frac{a}{z} - \frac{x}{c} = \frac{ac - xz}{cz}
$$

Now divide:
$$
\frac{(ay + bx)/by}{(ac - xz)/cz} = \frac{ay + bx}{by} \cdot \frac{cz}{ac - xz} = \frac{cz(ay + bx)}{by(ac - xz)}
$$

✔ Correct! Answer: $ \frac{cz(ay + bx)}{by(ac - xz)} $

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🔢 Problem 9:

$$
\frac{1 + \frac{x}{y}}{1 - \frac{x^2}{y^2}} = \frac{y(y - x)}{?}
$$

Wait — the given answer is only $ y(y - x) $, which is incomplete.

Let’s simplify:

Numerator: $ 1 + \frac{x}{y} = \frac{y + x}{y} $

Denominator: $ 1 - \frac{x^2}{y^2} = \frac{y^2 - x^2}{y^2} = \frac{(y - x)(y + x)}{y^2} $

Now divide:
$$
\frac{(x+y)/y}{(y^2 - x^2)/y^2} = \frac{x+y}{y} \cdot \frac{y^2}{(y-x)(y+x)} = \frac{y}{y - x}
$$

So the simplified form is $ \frac{y}{y - x} $

But the answer says $ y(y - x) $ — that's wrong.

✔ Correct Answer: $ \frac{y}{y - x} $

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🔢 Problem 10:

$$
\frac{x + \frac{a}{c}}{x + \frac{b}{d}} = \frac{d(a + cx)}{c(b + dx)}
$$

Simplify numerator: $ x + \frac{a}{c} = \frac{cx + a}{c} $

Denominator: $ x + \frac{b}{d} = \frac{dx + b}{d} $

Divide:
$$
\frac{(cx + a)/c}{(dx + b)/d} = \frac{cx + a}{c} \cdot \frac{d}{dx + b} = \frac{d(cx + a)}{c(dx + b)} = \frac{d(a + cx)}{c(b + dx)}
$$

✔ Correct! Answer: $ \frac{d(a + cx)}{c(b + dx)} $

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🔢 Problem 11:

$$
\frac{a^2 + \frac{x}{3}}{4 + \frac{5}{x}} = \frac{3(3a^2 + x)}{3(20 + x)}
$$

Wait — let's check.

Numerator: $ a^2 + \frac{x}{3} = \frac{3a^2 + x}{3} $

Denominator: $ 4 + \frac{5}{x} = \frac{4x + 5}{x} $

Now divide:
$$
\frac{(3a^2 + x)/3}{(4x + 5)/x} = \frac{3a^2 + x}{3} \cdot \frac{x}{4x + 5} = \frac{x(3a^2 + x)}{3(4x + 5)}
$$

But the given answer is $ \frac{3(3a^2 + x)}{3(20 + x)} = \frac{3a^2 + x}{20 + x} $ — which is incorrect

✔ Correct Answer: $ \frac{x(3a^2 + x)}{3(4x + 5)} $

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🔢 Problem 12:

$$
\frac{3a^2 - 3y^2}{a + y} = 9(a - y)
$$

Wait — numerator: $ 3a^2 - 3y^2 = 3(a^2 - y^2) = 3(a - y)(a + y) $

So:
$$
\frac{3(a - y)(a + y)}{a + y} = 3(a - y)
$$

But the answer says $ 9(a - y) $ → Incorrect

✔ Correct Answer: $ 3(a - y) $

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🔢 Problem 13:

$$
\frac{x + \frac{2d}{3ac}}{x + \frac{3d}{2ac}} = \frac{2(3acx + 2d)}{3(2acx + 3d)}
$$

Simplify numerator: $ x + \frac{2d}{3ac} = \frac{3acx + 2d}{3ac} $

Denominator: $ x + \frac{3d}{2ac} = \frac{2acx + 3d}{2ac} $

Now divide:
$$
\frac{(3acx + 2d)/(3ac)}{(2acx + 3d)/(2ac)} = \frac{3acx + 2d}{3ac} \cdot \frac{2ac}{2acx + 3d} = \frac{2(3acx + 2d)}{3(2acx + 3d)}
$$

✔ Correct! Answer: $ \frac{2(3acx + 2d)}{3(2acx + 3d)} $

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🔢 Problem 14:

$$
\frac{4a^2 - 4x^2}{\frac{a + x}{a - x}} = 4(a - x)^2
$$

First: $ 4a^2 - 4x^2 = 4(a^2 - x^2) = 4(a - x)(a + x) $

Now divide:
$$
\frac{4(a - x)(a + x)}{(a + x)/(a - x)} = 4(a - x)(a + x) \cdot \frac{a - x}{a + x} = 4(a - x)^2
$$

✔ Correct! Answer: $ 4(a - x)^2 $

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## ✔ Summary of Corrections:

| Problem | Given Answer | Correct Answer |
|--------|---------------|----------------|
| 1 | $ \frac{85}{12} $ | $ \frac{17}{12} $ |
| 2 | $ \frac{5}{8} $ | $ \frac{5}{12} $ |
| 3 | $ \frac{65}{132} $ | $ \frac{13}{12} $ |
| 4 | $ \frac{12}{13} $ | ✔ Correct |
| 5 | $ \frac{69}{95} $ | ✔ Correct |
| 6 | $ \frac{101}{92} $ | $ \frac{43}{30} $ |
| 7 | $ \frac{3(4x+y)}{4(3x-y)} $ | ✔ Correct |
| 8 | $ \frac{cz(ay+bx)}{by(ac-xz)} $ | ✔ Correct |
| 9 | $ y(y - x) $ | $ \frac{y}{y - x} $ |
| 10 | $ \frac{d(a+cx)}{c(b+dx)} $ | ✔ Correct |
| 11 | $ \frac{3(3a^2+x)}{3(20+x)} $ | $ \frac{x(3a^2 + x)}{3(4x + 5)} $ |
| 12 | $ 9(a - y) $ | $ 3(a - y) $ |
| 13 | $ \frac{2(3acx+2d)}{3(2acx+3d)} $ | ✔ Correct |
| 14 | $ 4(a - x)^2 $ | ✔ Correct |

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## 📌 Final Notes:
- Problems 4, 5, 7, 8, 10, 13, 14 are correctly solved.
- The rest have errors in the provided answers.
- Common mistakes: incorrect LCDs, algebraic missteps, or missing simplifications.

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