Find the unknown number in the proportion

if a/9 = b/2 then (a + 9) / 9 = x/ y
 
 
 


We are given two ratios:  
\[
\frac{a}{9} = \frac{b}{2}
\]
and 
\[
\frac{a + 9}{9} = \frac{x}{y}.
\]

### Step 1: Solve for \( a \) in terms of \( b \)
Using the first equation, cross-multiply to find the relationship between \( a \) and \( b \):

\[
a \cdot 2 = b \cdot 9
\]
\[
2a = 9b
\]
\[
a = \frac{9b}{2}.
\]

### Step 2: Substitute \( a \) into the second equation
Now, substitute \( a = \frac{9b}{2} \) into the second equation:

\[
\frac{\left(\frac{9b}{2} + 9\right)}{9} = \frac{x}{y}.
\]

### Step 3: Simplify the numerator
Simplify \( \frac{9b}{2} + 9 \):

\[
\frac{9b}{2} + 9 = \frac{9b}{2} + \frac{18}{2} = \frac{9b + 18}{2}.
\]

Thus, the second equation becomes:
\[
\frac{\frac{9b + 18}{2}}{9} = \frac{x}{y}.
\]

### Step 4: Simplify the left-hand side
Simplify the expression \( \frac{\frac{9b + 18}{2}}{9} \):
\[
\frac{\frac{9b + 18}{2}}{9} = \frac{9b + 18}{18}.
\]

So the equation becomes:
\[
\frac{9b + 18}{18} = \frac{x}{y}.
\]

### Step 5: Compare numerators and denominators
At this point, we have:
\[
\frac{9b + 18}{18} = \frac{x}{y}.
\]

Therefore, the unknown numbers \( x \) and \( y \) must satisfy the following:

\[
x = 9b + 18 \quad \text{and} \quad y = 18.
\]

Thus, the values of \( x \) and \( y \) are:
\[
x = 9b + 18, \quad y = 18.
\]

Let me know if you need further details or clarifications!

---

Here are five related questions:
1. How do you cross-multiply proportions?
2. What is the significance of simplifying equations in proportional problems?
3. Can we express \( b \) in terms of \( a \) and solve the problem differently?
4. What happens if \( b \) takes specific values, like 1 or 2?
5. How can you check if a proportion holds true for given numbers?

**Tip:** In a proportion, cross-multiplication is a quick way to compare the equality of two ratios.

Find the unknown number in the proportion if a/9 = b/2 then (a + 9) / 9 = x/ y

Learn how to solve a proportion where a/9 = b/2 and (a + 9) / 9 = x / y. This detailed step-by-step solution covers algebraic methods and cross-multiplication. Suitable for Grades 8-10 students learning about proportions and algebra.

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