Solving for ( frac{a}{b} ) Given ( a times b 76 ) and ( a - b 38 )

In this article, we will explore the process of solving for ( frac{a}{b} ) using the given equations ( a times b 76 ) and ( a - b 38 ). By solving these equations step-by-step, we aim to provide a clear understanding of the algebra involved in such problem-solving exercises.

Solving the System of Equations

To find the values of ( a ) and ( b ), we start by adding the two given equations:

Step 1: Adding Equations

Given:

Equation 1: ( a times b 76 ) Equation 2: ( a - b 38 )

Adding these two equations, we get:

( a times b a - b 76 38 )

This simplifies to:

( 2a 114 )

Step 2: Solving for ( a )

Dividing both sides by 2, we obtain:

( a 57 )

Step 3: Substituting ( a 57 ) Back into One of the Original Equations

Now, substituting ( a 57 ) into Equation 1:

( 57 times b 76 )

Solving for ( b ), we get:

( b frac{76}{57} frac{76 - 57}{1} 19 )

Step 4: Calculating ( frac{a}{b} )

Now that we have ( a 57 ) and ( b 19 ), we can find ( frac{a}{b} ):

( frac{a}{b} frac{57}{19} )

This simplifies to:

( frac{57}{19} 3 )

Thus, when you divide ( a ) by ( b ), you get 3.

Alternative Method: Using Componendo and Dividendo

Another way to solve this problem is by using the properties of componendo and dividendo. Here are the steps:

Step 1: Adding and Subtracting the Equations

Given:

Equation 1: ( a times b 76 ) Equation 2: ( a - b 38 )

Using these, we can write:

( frac{a b}{a - b} frac{76 38}{76 - 38} )

This simplifies to:

( frac{a b}{a - b} frac{114}{38} )

Step 2: Simplifying Further

( frac{a b}{a - b} 3 )

Step 3: Using Componendo and Dividendo

Applying the property ( frac{a b}{a - b} 3 ), we can write:

( frac{2a}{2b} 3 )

Which simplifies to:

( frac{a}{b} 3 )

This confirms our previous result.

Conclusion

The value of ( frac{a}{b} ) when given the equations ( a times b 76 ) and ( a - b 38 ) is 3. This solution demonstrates the application of basic algebraic techniques to solve systems of equations and the use of componendo and dividendo for simplification. By understanding these methods, you can solve similar problems more effectively.

Related Keywords: equation solving, system of equations, algebraic equations