Math Exercises & Math Problems: Algebraic Fractions - Free Printable
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Step-by-step solution for: Math Exercises & Math Problems: Algebraic Fractions
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Show Answer Key & Explanations
Step-by-step solution for: Math Exercises & Math Problems: Algebraic Fractions
To simplify the given expressions, we will use the principles of fraction simplification, which involve canceling out common factors in the numerator and the denominator. Let's go through each expression step by step.
---
- The numerator is $3$, and the denominator is $12$.
- The greatest common divisor (GCD) of $3$ and $12$ is $3$.
- Divide both the numerator and the denominator by $3$:
$$
\frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4}
$$
Answer: $\boxed{\frac{1}{4}}$
---
- The numerator is $3x$, and the denominator is $12$.
- The GCD of $3$ and $12$ is $3$.
- Divide both the numerator and the denominator by $3$:
$$
\frac{3x}{12} = \frac{3x \div 3}{12 \div 3} = \frac{x}{4}
$$
Answer: $\boxed{\frac{x}{4}}$
---
- The numerator is $3$, and the denominator is $12x$.
- The GCD of $3$ and $12$ is $3$.
- Divide both the numerator and the coefficient of $x$ in the denominator by $3$:
$$
\frac{3}{12x} = \frac{3 \div 3}{12 \div 3 \cdot x} = \frac{1}{4x}
$$
Answer: $\boxed{\frac{1}{4x}}$
---
- The numerator is $12x$, and the denominator is $3$.
- The GCD of $12$ and $3$ is $3$.
- Divide both the numerator and the denominator by $3$:
$$
\frac{12x}{3} = \frac{12 \div 3 \cdot x}{3 \div 3} = \frac{4x}{1} = 4x
$$
Answer: $\boxed{4x}$
---
- The numerator is $12x$, and the denominator is $6$.
- The GCD of $12$ and $6$ is $6$.
- Divide both the numerator and the denominator by $6$:
$$
\frac{12x}{6} = \frac{12 \div 6 \cdot x}{6 \div 6} = \frac{2x}{1} = 2x
$$
Answer: $\boxed{2x}$
---
- The numerator is $12x$, and the denominator is $6x$.
- Both the numerator and the denominator have a common factor of $6x$.
- Divide both the numerator and the denominator by $6x$:
$$
\frac{12x}{6x} = \frac{12 \div 6 \cdot x \div x}{6 \div 6 \cdot x \div x} = \frac{2}{1} = 2
$$
Answer: $\boxed{2}$
---
- The numerator is $12xy$, and the denominator is $6x$.
- Both the numerator and the denominator have a common factor of $6x$.
- Divide both the numerator and the denominator by $6x$:
$$
\frac{12xy}{6x} = \frac{12 \div 6 \cdot x \div x \cdot y}{6 \div 6 \cdot x \div x} = \frac{2y}{1} = 2y
$$
Answer: $\boxed{2y}$
---
- The numerator is $6x$, and the denominator is $12xy$.
- Both the numerator and the denominator have a common factor of $6x$.
- Divide both the numerator and the denominator by $6x$:
$$
\frac{6x}{12xy} = \frac{6 \div 6 \cdot x \div x}{12 \div 6 \cdot x \div x \cdot y} = \frac{1}{2y}
$$
Answer: $\boxed{\frac{1}{2y}}$
---
- The numerator is $10x$, and the denominator is $12xy$.
- Both the numerator and the denominator have a common factor of $2x$.
- Divide both the numerator and the denominator by $2x$:
$$
\frac{10x}{12xy} = \frac{10 \div 2 \cdot x \div x}{12 \div 2 \cdot x \div x \cdot y} = \frac{5}{6y}
$$
Answer: $\boxed{\frac{5}{6y}}$
---
- The numerator is $10x$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2x$.
- Divide both the numerator and the denominator by $2x$:
$$
\frac{10x}{12xy^2} = \frac{10 \div 2 \cdot x \div x}{12 \div 2 \cdot x \div x \cdot y^2} = \frac{5}{6y^2}
$$
Answer: $\boxed{\frac{5}{6y^2}}$
---
- The numerator is $10xy$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy$.
- Divide both the numerator and the denominator by $2xy$:
$$
\frac{10xy}{12xy^2} = \frac{10 \div 2 \cdot x \div x \cdot y \div y}{12 \div 2 \cdot x \div x \cdot y \div y \cdot y} = \frac{5}{6y}
$$
Answer: $\boxed{\frac{5}{6y}}$
---
- The numerator is $10x^2y$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy$.
- Divide both the numerator and the denominator by $2xy$:
$$
\frac{10x^2y}{12xy^2} = \frac{10 \div 2 \cdot x^2 \div x \cdot y \div y}{12 \div 2 \cdot x \div x \cdot y^2 \div y} = \frac{5x}{6y}
$$
Answer: $\boxed{\frac{5x}{6y}}$
---
- The numerator is $10x^2y^2$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy^2$.
- Divide both the numerator and the denominator by $2xy^2$:
$$
\frac{10x^2y^2}{12xy^2} = \frac{10 \div 2 \cdot x^2 \div x \cdot y^2 \div y^2}{12 \div 2 \cdot x \div x \cdot y^2 \div y^2} = \frac{5x}{6}
$$
Answer: $\boxed{\frac{5x}{6}}$
---
- The numerator is $10x^2y^3$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy^2$.
- Divide both the numerator and the denominator by $2xy^2$:
$$
\frac{10x^2y^3}{12xy^2} = \frac{10 \div 2 \cdot x^2 \div x \cdot y^3 \div y^2}{12 \div 2 \cdot x \div x \cdot y^2 \div y^2} = \frac{5xy}{6}
$$
Answer: $\boxed{\frac{5xy}{6}}$
---
- The numerator is $100x^2y^4$, and the denominator is $120xy^2$.
- Both the numerator and the denominator have a common factor of $20xy^2$.
- Divide both the numerator and the denominator by $20xy^2$:
$$
\frac{100x^2y^4}{120xy^2} = \frac{100 \div 20 \cdot x^2 \div x \cdot y^4 \div y^2}{120 \div 20 \cdot x \div x \cdot y^2 \div y^2} = \frac{5xy^2}{6}
$$
Answer: $\boxed{\frac{5xy^2}{6}}$
---
- The numerator is $100x^2y^4$, and the denominator is $xy^2$.
- Both the numerator and the denominator have a common factor of $xy^2$.
- Divide both the numerator and the denominator by $xy^2$:
$$
\frac{100x^2y^4}{xy^2} = \frac{100 \cdot x^2 \div x \cdot y^4 \div y^2}{x \div x \cdot y^2 \div y^2} = \frac{100xy^2}{1} = 100xy^2
$$
Answer: $\boxed{100xy^2}$
---
\[
\boxed{
\begin{array}{ll}
(a) & \frac{1}{4} \\
(b) & \frac{x}{4} \\
(c) & \frac{1}{4x} \\
(d) & 4x \\
(e) & 2x \\
(f) & 2 \\
(g) & 2y \\
(h) & \frac{1}{2y} \\
(i) & \frac{5}{6y} \\
(j) & \frac{5}{6y^2} \\
(k) & \frac{5}{6y} \\
(l) & \frac{5x}{6y} \\
(m) & \frac{5x}{6} \\
(n) & \frac{5xy}{6} \\
(o) & \frac{5xy^2}{6} \\
(p) & 100xy^2 \\
\end{array}
}
\]
---
(a) Simplify: $\frac{3}{12}$
- The numerator is $3$, and the denominator is $12$.
- The greatest common divisor (GCD) of $3$ and $12$ is $3$.
- Divide both the numerator and the denominator by $3$:
$$
\frac{3}{12} = \frac{3 \div 3}{12 \div 3} = \frac{1}{4}
$$
Answer: $\boxed{\frac{1}{4}}$
---
(b) Simplify: $\frac{3x}{12}$
- The numerator is $3x$, and the denominator is $12$.
- The GCD of $3$ and $12$ is $3$.
- Divide both the numerator and the denominator by $3$:
$$
\frac{3x}{12} = \frac{3x \div 3}{12 \div 3} = \frac{x}{4}
$$
Answer: $\boxed{\frac{x}{4}}$
---
(c) Simplify: $\frac{3}{12x}$
- The numerator is $3$, and the denominator is $12x$.
- The GCD of $3$ and $12$ is $3$.
- Divide both the numerator and the coefficient of $x$ in the denominator by $3$:
$$
\frac{3}{12x} = \frac{3 \div 3}{12 \div 3 \cdot x} = \frac{1}{4x}
$$
Answer: $\boxed{\frac{1}{4x}}$
---
(d) Simplify: $\frac{12x}{3}$
- The numerator is $12x$, and the denominator is $3$.
- The GCD of $12$ and $3$ is $3$.
- Divide both the numerator and the denominator by $3$:
$$
\frac{12x}{3} = \frac{12 \div 3 \cdot x}{3 \div 3} = \frac{4x}{1} = 4x
$$
Answer: $\boxed{4x}$
---
(e) Simplify: $\frac{12x}{6}$
- The numerator is $12x$, and the denominator is $6$.
- The GCD of $12$ and $6$ is $6$.
- Divide both the numerator and the denominator by $6$:
$$
\frac{12x}{6} = \frac{12 \div 6 \cdot x}{6 \div 6} = \frac{2x}{1} = 2x
$$
Answer: $\boxed{2x}$
---
(f) Simplify: $\frac{12x}{6x}$
- The numerator is $12x$, and the denominator is $6x$.
- Both the numerator and the denominator have a common factor of $6x$.
- Divide both the numerator and the denominator by $6x$:
$$
\frac{12x}{6x} = \frac{12 \div 6 \cdot x \div x}{6 \div 6 \cdot x \div x} = \frac{2}{1} = 2
$$
Answer: $\boxed{2}$
---
(g) Simplify: $\frac{12xy}{6x}$
- The numerator is $12xy$, and the denominator is $6x$.
- Both the numerator and the denominator have a common factor of $6x$.
- Divide both the numerator and the denominator by $6x$:
$$
\frac{12xy}{6x} = \frac{12 \div 6 \cdot x \div x \cdot y}{6 \div 6 \cdot x \div x} = \frac{2y}{1} = 2y
$$
Answer: $\boxed{2y}$
---
(h) Simplify: $\frac{6x}{12xy}$
- The numerator is $6x$, and the denominator is $12xy$.
- Both the numerator and the denominator have a common factor of $6x$.
- Divide both the numerator and the denominator by $6x$:
$$
\frac{6x}{12xy} = \frac{6 \div 6 \cdot x \div x}{12 \div 6 \cdot x \div x \cdot y} = \frac{1}{2y}
$$
Answer: $\boxed{\frac{1}{2y}}$
---
(i) Simplify: $\frac{10x}{12xy}$
- The numerator is $10x$, and the denominator is $12xy$.
- Both the numerator and the denominator have a common factor of $2x$.
- Divide both the numerator and the denominator by $2x$:
$$
\frac{10x}{12xy} = \frac{10 \div 2 \cdot x \div x}{12 \div 2 \cdot x \div x \cdot y} = \frac{5}{6y}
$$
Answer: $\boxed{\frac{5}{6y}}$
---
(j) Simplify: $\frac{10x}{12xy^2}$
- The numerator is $10x$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2x$.
- Divide both the numerator and the denominator by $2x$:
$$
\frac{10x}{12xy^2} = \frac{10 \div 2 \cdot x \div x}{12 \div 2 \cdot x \div x \cdot y^2} = \frac{5}{6y^2}
$$
Answer: $\boxed{\frac{5}{6y^2}}$
---
(k) Simplify: $\frac{10xy}{12xy^2}$
- The numerator is $10xy$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy$.
- Divide both the numerator and the denominator by $2xy$:
$$
\frac{10xy}{12xy^2} = \frac{10 \div 2 \cdot x \div x \cdot y \div y}{12 \div 2 \cdot x \div x \cdot y \div y \cdot y} = \frac{5}{6y}
$$
Answer: $\boxed{\frac{5}{6y}}$
---
(l) Simplify: $\frac{10x^2y}{12xy^2}$
- The numerator is $10x^2y$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy$.
- Divide both the numerator and the denominator by $2xy$:
$$
\frac{10x^2y}{12xy^2} = \frac{10 \div 2 \cdot x^2 \div x \cdot y \div y}{12 \div 2 \cdot x \div x \cdot y^2 \div y} = \frac{5x}{6y}
$$
Answer: $\boxed{\frac{5x}{6y}}$
---
(m) Simplify: $\frac{10x^2y^2}{12xy^2}$
- The numerator is $10x^2y^2$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy^2$.
- Divide both the numerator and the denominator by $2xy^2$:
$$
\frac{10x^2y^2}{12xy^2} = \frac{10 \div 2 \cdot x^2 \div x \cdot y^2 \div y^2}{12 \div 2 \cdot x \div x \cdot y^2 \div y^2} = \frac{5x}{6}
$$
Answer: $\boxed{\frac{5x}{6}}$
---
(n) Simplify: $\frac{10x^2y^3}{12xy^2}$
- The numerator is $10x^2y^3$, and the denominator is $12xy^2$.
- Both the numerator and the denominator have a common factor of $2xy^2$.
- Divide both the numerator and the denominator by $2xy^2$:
$$
\frac{10x^2y^3}{12xy^2} = \frac{10 \div 2 \cdot x^2 \div x \cdot y^3 \div y^2}{12 \div 2 \cdot x \div x \cdot y^2 \div y^2} = \frac{5xy}{6}
$$
Answer: $\boxed{\frac{5xy}{6}}$
---
(o) Simplify: $\frac{100x^2y^4}{120xy^2}$
- The numerator is $100x^2y^4$, and the denominator is $120xy^2$.
- Both the numerator and the denominator have a common factor of $20xy^2$.
- Divide both the numerator and the denominator by $20xy^2$:
$$
\frac{100x^2y^4}{120xy^2} = \frac{100 \div 20 \cdot x^2 \div x \cdot y^4 \div y^2}{120 \div 20 \cdot x \div x \cdot y^2 \div y^2} = \frac{5xy^2}{6}
$$
Answer: $\boxed{\frac{5xy^2}{6}}$
---
(p) Simplify: $\frac{100x^2y^4}{xy^2}$
- The numerator is $100x^2y^4$, and the denominator is $xy^2$.
- Both the numerator and the denominator have a common factor of $xy^2$.
- Divide both the numerator and the denominator by $xy^2$:
$$
\frac{100x^2y^4}{xy^2} = \frac{100 \cdot x^2 \div x \cdot y^4 \div y^2}{x \div x \cdot y^2 \div y^2} = \frac{100xy^2}{1} = 100xy^2
$$
Answer: $\boxed{100xy^2}$
---
Final Answers:
\[
\boxed{
\begin{array}{ll}
(a) & \frac{1}{4} \\
(b) & \frac{x}{4} \\
(c) & \frac{1}{4x} \\
(d) & 4x \\
(e) & 2x \\
(f) & 2 \\
(g) & 2y \\
(h) & \frac{1}{2y} \\
(i) & \frac{5}{6y} \\
(j) & \frac{5}{6y^2} \\
(k) & \frac{5}{6y} \\
(l) & \frac{5x}{6y} \\
(m) & \frac{5x}{6} \\
(n) & \frac{5xy}{6} \\
(o) & \frac{5xy^2}{6} \\
(p) & 100xy^2 \\
\end{array}
}
\]
Parent Tip: Review the logic above to help your child master the concept of simplifying algebraic fractions worksheet.
