Solving the Ratio Equation: (9g + 2h) : (9g - 2h) = 9 : 2

This problem asks us to find the ratio of two variables, g and h, given an equation involving a ratio of expressions containing g and h. We are given the ratio equation:

$$(9g + 2h) : (9g - 2h) = 9 : 2$$

To solve this ratio equation and find the value of g : h, we can convert the ratio notation into a fractional equation. A ratio $a : b$ is equivalent to the fraction $\frac{a}{b}$. Applying this to our given equation, we get:

$$\frac{9g + 2h}{9g - 2h} = \frac{9}{2}$$

Step-by-Step Solution to Find g : h

Now we have a standard algebraic equation with fractions. To eliminate the denominators, we can perform cross-multiplication. Cross-multiplication involves multiplying the numerator of one side by the denominator of the other side and setting the results equal.

Multiplying the numerator of the left side by the denominator of the right side, and the numerator of the right side by the denominator of the left side, we get:

$$2 \times (9g + 2h) = 9 \times (9g - 2h)$$

Next, we distribute the numbers outside the parentheses to the terms inside the parentheses:

$$2 \times 9g + 2 \times 2h = 9 \times 9g - 9 \times 2h$$

This simplifies to:

$$18g + 4h = 81g - 18h$$

Our goal is to find the ratio g : h, which means we want to express $\frac{g}{h}$ or $\frac{h}{g}$. To do this, we need to gather all the terms containing 'g' on one side of the equation and all the terms containing 'h' on the other side.

Let's move the $18g$ term from the left side to the right side by subtracting $18g$ from both sides:

$$4h = 81g - 18h - 18g$$

$$4h = 63g - 18h$$

Now, let's move the $-18h$ term from the right side to the left side by adding $18h$ to both sides:

$$4h + 18h = 63g$$

Combine the like terms on the left side:

$$22h = 63g$$

Finally, to find the ratio $g : h$ (or $\frac{g}{h}$), we can rearrange the equation. We want to isolate $\frac{g}{h}$. Divide both sides by $63h$ (assuming $h \neq 0$ and $g \neq 0$, which must be true for the original ratio to be defined and finite):

$$\frac{22h}{63h} = \frac{63g}{63h}$$

Cancel out the common terms:

$$\frac{22}{63} = \frac{g}{h}$$

So, the ratio $\frac{g}{h}$ is $\frac{22}{63}$. This means $g : h = 22 : 63$.

Verification of the Solution

We can check if our ratio $g:h = 22:63$ satisfies the original equation. Let $g = 22k$ and $h = 63k$ for some non-zero constant $k$. Substitute these values into the original ratio $(9g + 2h) : (9g - 2h)$:

Numerator: $9g + 2h = 9(22k) + 2(63k) = 198k + 126k = 324k$

Denominator: $9g - 2h = 9(22k) - 2(63k) = 198k - 126k = 72k$

The ratio is $\frac{324k}{72k} = \frac{324}{72}$.

Simplify the fraction $\frac{324}{72}$ by dividing both numerator and denominator by common factors. Both are divisible by 9:

$\frac{324 \div 9}{72 \div 9} = \frac{36}{8}$

Both are divisible by 4:

$\frac{36 \div 4}{8 \div 4} = \frac{9}{2}$

The ratio is $\frac{9}{2}$, which is $9 : 2$. This matches the given ratio in the question, confirming our solution is correct.

Revision Table: Key Concepts for Ratio Problems

Additional Information: Ratios and Proportions

A ratio is a way to compare the relative sizes of two or more quantities. For example, if a class has 10 boys and 15 girls, the ratio of boys to girls is $10:15$, which simplifies to $2:3$. This means for every 2 boys, there are 3 girls.

A proportion is an equation stating that two ratios are equal. The equation $\frac{a}{b} = \frac{c}{d}$ is a proportion. The problem we solved is essentially a problem involving a proportion derived from expressions containing variables.

Solving ratio and proportion problems often involves converting ratios to fractions and using algebraic techniques like cross-multiplication and rearranging terms to find unknown values or ratios.

Another method to solve $\frac{9g + 2h}{9g - 2h} = \frac{9}{2}$ is using the Componendo and Dividendo rule, which states that if $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$.

Applying Componendo and Dividendo to $\frac{9g + 2h}{9g - 2h} = \frac{9}{2}$:

Let $a = 9g + 2h$ and $b = 9g - 2h$. Let $c = 9$ and $d = 2$.

$$\frac{(9g + 2h) + (9g - 2h)}{(9g + 2h) - (9g - 2h)} = \frac{9 + 2}{9 - 2}$$

Simplify the numerator and denominator on the left side:

Numerator: $(9g + 2h) + (9g - 2h) = 9g + 2h + 9g - 2h = 18g$

Denominator: $(9g + 2h) - (9g - 2h) = 9g + 2h - 9g + 2h = 4h$ (Note the sign change for the terms being subtracted).

Simplify the right side:

Numerator: $9 + 2 = 11$

Denominator: $9 - 2 = 7$

Substituting these back into the Componendo and Dividendo formula:

$$\frac{18g}{4h} = \frac{11}{7}$$

Simplify the fraction on the left side:

$$\frac{9g}{2h} = \frac{11}{7}$$

Now, isolate $\frac{g}{h}$. Multiply both sides by $\frac{2}{9}$:

$$\frac{9g}{2h} \times \frac{2}{9} = \frac{11}{7} \times \frac{2}{9}$$

$$\frac{g}{h} = \frac{11 \times 2}{7 \times 9}$$

$$\frac{g}{h} = \frac{22}{63}$$

This confirms our previous result using a different method. The ratio $g : h$ is $22 : 63$.