What number must be added to each term of the ratio `9 : 16` to make the ratio `2 : 3`

Answer

To solve the problem of what number must be added to each term of the ratio \(9 : 16\) to make the ratio \(2 : 3\), we can follow these steps:

### Step 1: Define the unknown
Let the number that we need to add to each term of the ratio \(9 : 16\) be \(x\).

### Step 2: Set up the equation
After adding \(x\) to each term, the new ratio becomes:
\[ \frac{9 + x}{16 + x} = \frac{2}{3} \]

### Step 3: Cross-multiply to eliminate the fraction
Cross-multiplying gives us:
\[ 3(9 + x) = 2(16 + x) \]

### Step 4: Expand both sides
Expanding both sides of the equation results in:
\[ 27 + 3x = 32 + 2x \]

### Step 5: Rearrange the equation
To isolate \(x\), we can subtract \(2x\) from both sides:
\[ 27 + 3x - 2x = 32 \]
This simplifies to:
\[ 27 + x = 32 \]

### Step 6: Solve for \(x\)
Now, subtract \(27\) from both sides:
\[ x = 32 - 27 \]
Thus, we find:
\[ x = 5 \]

### Conclusion
The number that must be added to each term of the ratio \(9 : 16\) to make the ratio \(2 : 3\) is \(5\).

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