Solving the Ratio Equation: If 5a : 3b 2a - 3b 23 : 5 Then What is the Value of a : b

In this article, we will solve a complex ratio equation provided by the problem:

The Given Problem and Solution Process

The given problem is as follows:

5a : 3b 23 : 5 2a - 3b 23 : 5

We need to determine the value of a : b.

Step-by-Step Solution

Step 1: Understanding the First Equation

The first equation is given as:

5a : 3b 23 : 5

This can be written as a proportion:

[[frac{5a}{3b} frac{23}{5}]

Multiplying both sides by 3b, we get:

[[5a frac{23 times 3b}{5}]

Which simplifies to:

[[5a frac{69b}{5}]

Multiplying both sides by 5, we get:

[[25a 69b]]

Rearranging this equation, we have:

[[25a - 69b 0]

Step 2: Understanding the Second Equation

The second equation is:

2a - 3b 23 : 5

This can be written as:

[[2a - 3b 5]

Step 3: Solving the System of Equations

We now have two equations:

[[begin{cases} 25a - 69b 0 2a - 3b 5 end{cases}]

We can solve this system of linear equations to find the values of a and b.

Step 4: Elimination Method

First, we multiply the second equation by 25 to eliminate a:

[[50a - 75b 125]

Next, we multiply the first equation by 2 to make the coefficients of a the same:

[[50a - 138b 0]

Subtracting the second equation from the first:

[[50a - 138b 0] - [50a - 75b 125]

This simplifies to:

[[-63b 125]

Therefore:

[[b -frac{125}{63}]

Substituting b back into one of the original equations, we can find a:

[[2a - 3left(-frac{125}{63}right) 5]

[2a frac{375}{63} 5]

[2a 5 - frac{375}{63}]

[[2a frac{315 - 375}{63}]]

[[2a -frac{60}{63}]

[[a -frac{30}{63}]

[[a -frac{10}{21}]

Step 5: Finding the Ratio a : b

The values we found are:

[[a -frac{10}{21}, b -frac{125}{63}]

The ratio a : b is:

[[frac{-frac{10}{21}}{-frac{125}{63}} frac{10 times 63}{21 times 125} frac{630}{2625} frac{2}{5}]

Thus, the value of a : b is 4 : 1.

Conclusion

The value of a : b is 4 : 1. This conclusion was reached through solving the system of equations and finding the ratio.

Additional Notes

The problem involves understanding and applying the concept of ratios and solving systems of linear equations. This is an important skill in algebra and can be useful in various real-world applications such as finance, physics, and engineering.

Keywords: ratio equation, algebraic ratio, solving ratios