Basic Math Examples

$\frac{3 a - 1}{4} + \frac{a - 2}{3} - \frac{a - 1}{5} = \frac{21}{20}$

Step 1

Simplify $\frac{3 a - 1}{4} + \frac{a - 2}{3} - \frac{a - 1}{5}$ .

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Step 1.1

Find the common denominator.

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Step 1.1.1

Multiply $\frac{3 a - 1}{4}$ by $\frac{15}{15}$ .

$\frac{3 a - 1}{4} \cdot \frac{15}{15} + \frac{a - 2}{3} - \frac{a - 1}{5} = \frac{21}{20}$

Step 1.1.2

Multiply $\frac{3 a - 1}{4}$ by $\frac{15}{15}$ .

$\frac{(3 a - 1) \cdot 15}{4 \cdot 15} + \frac{a - 2}{3} - \frac{a - 1}{5} = \frac{21}{20}$

Step 1.1.3

Multiply $\frac{a - 2}{3}$ by $\frac{20}{20}$ .

$\frac{(3 a - 1) \cdot 15}{4 \cdot 15} + \frac{a - 2}{3} \cdot \frac{20}{20} - \frac{a - 1}{5} = \frac{21}{20}$

Step 1.1.4

Multiply $\frac{a - 2}{3}$ by $\frac{20}{20}$ .

$\frac{(3 a - 1) \cdot 15}{4 \cdot 15} + \frac{(a - 2) \cdot 20}{3 \cdot 20} - \frac{a - 1}{5} = \frac{21}{20}$

Step 1.1.5

Multiply $\frac{a - 1}{5}$ by $\frac{12}{12}$ .

$\frac{(3 a - 1) \cdot 15}{4 \cdot 15} + \frac{(a - 2) \cdot 20}{3 \cdot 20} - (\frac{a - 1}{5} \cdot \frac{12}{12}) = \frac{21}{20}$

Step 1.1.6

Multiply $\frac{a - 1}{5}$ by $\frac{12}{12}$ .

$\frac{(3 a - 1) \cdot 15}{4 \cdot 15} + \frac{(a - 2) \cdot 20}{3 \cdot 20} - \frac{(a - 1) \cdot 12}{5 \cdot 12} = \frac{21}{20}$

Step 1.1.7

Multiply $4$ by $15$ .

$\frac{(3 a - 1) \cdot 15}{60} + \frac{(a - 2) \cdot 20}{3 \cdot 20} - \frac{(a - 1) \cdot 12}{5 \cdot 12} = \frac{21}{20}$

Step 1.1.8

Multiply $3$ by $20$ .

$\frac{(3 a - 1) \cdot 15}{60} + \frac{(a - 2) \cdot 20}{60} - \frac{(a - 1) \cdot 12}{5 \cdot 12} = \frac{21}{20}$

Step 1.1.9

Multiply $5$ by $12$ .

$\frac{(3 a - 1) \cdot 15}{60} + \frac{(a - 2) \cdot 20}{60} - \frac{(a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.2

Combine the numerators over the common denominator.

$\frac{(3 a - 1) \cdot 15 + (a - 2) \cdot 20 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3

Simplify each term.

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Step 1.3.1

Apply the distributive property.

$\frac{3 a \cdot 15 - 1 \cdot 15 + (a - 2) \cdot 20 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.2

Multiply $15$ by $3$ .

$\frac{45 a - 1 \cdot 15 + (a - 2) \cdot 20 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.3

Multiply $- 1$ by $15$ .

$\frac{45 a - 15 + (a - 2) \cdot 20 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.4

Apply the distributive property.

$\frac{45 a - 15 + a \cdot 20 - 2 \cdot 20 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.5

Move $20$ to the left of $a$ .

$\frac{45 a - 15 + 20 \cdot a - 2 \cdot 20 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.6

Multiply $- 2$ by $20$ .

$\frac{45 a - 15 + 20 \cdot a - 40 - (a - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.7

Apply the distributive property.

$\frac{45 a - 15 + 20 a - 40 + (- a - - 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.8

Multiply $- 1$ by $- 1$ .

$\frac{45 a - 15 + 20 a - 40 + (- a + 1) \cdot 12}{60} = \frac{21}{20}$

Step 1.3.9

Apply the distributive property.

$\frac{45 a - 15 + 20 a - 40 - a \cdot 12 + 1 \cdot 12}{60} = \frac{21}{20}$

Step 1.3.10

Multiply $12$ by $- 1$ .

$\frac{45 a - 15 + 20 a - 40 - 12 a + 1 \cdot 12}{60} = \frac{21}{20}$

Step 1.3.11

Multiply $12$ by $1$ .

$\frac{45 a - 15 + 20 a - 40 - 12 a + 12}{60} = \frac{21}{20}$

Step 1.4

Simplify by adding terms.

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Step 1.4.1

Add $45 a$ and $20 a$ .

$\frac{65 a - 15 - 40 - 12 a + 12}{60} = \frac{21}{20}$

Step 1.4.2

Subtract $12 a$ from $65 a$ .

$\frac{53 a - 15 - 40 + 12}{60} = \frac{21}{20}$

Step 1.4.3

Simplify by adding and subtracting.

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Step 1.4.3.1

Subtract $40$ from $- 15$ .

$\frac{53 a - 55 + 12}{60} = \frac{21}{20}$

Step 1.4.3.2

Add $- 55$ and $12$ .

$\frac{53 a - 43}{60} = \frac{21}{20}$

Step 2

Multiply both sides by $60$ .

$\frac{53 a - 43}{60} \cdot 60 = \frac{21}{20} \cdot 60$

Step 3

Simplify.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Cancel the common factor of $60$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{53 a - 43}{60} \cdot 60 = \frac{21}{20} \cdot 60$

Step 3.1.1.2

Rewrite the expression.

$53 a - 43 = \frac{21}{20} \cdot 60$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{21}{20} \cdot 60$ .

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Step 3.2.1.1

Cancel the common factor of $20$ .

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Step 3.2.1.1.1

Factor $20$ out of $60$ .

$53 a - 43 = \frac{21}{20} \cdot (20 (3))$

Step 3.2.1.1.2

Cancel the common factor.

$53 a - 43 = \frac{21}{20} \cdot (20 \cdot 3)$

Step 3.2.1.1.3

Rewrite the expression.

$53 a - 43 = 21 \cdot 3$

Step 3.2.1.2

Multiply $21$ by $3$ .

$53 a - 43 = 63$

Step 4

Solve for $a$ .

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Step 4.1

Move all terms not containing $a$ to the right side of the equation.

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Step 4.1.1

Add $43$ to both sides of the equation.

$53 a = 63 + 43$

Step 4.1.2

Add $63$ and $43$ .

$53 a = 106$

Step 4.2

Divide each term in $53 a = 106$ by $53$ and simplify.

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Step 4.2.1

Divide each term in $53 a = 106$ by $53$ .

$\frac{53 a}{53} = \frac{106}{53}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $53$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{53 a}{53} = \frac{106}{53}$

Step 4.2.2.1.2

Divide $a$ by $1$ .

$a = \frac{106}{53}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $106$ by $53$ .

$a = 2$