Question

Which following equations are true for the number 9?

select all that apply.


___ 4 divide by ____ = 9/4

____ 1/9 = ____ divide by 9

____ 7 divide by _____ = 7/9

___ ___ divide by 1/9 = 81

Answer 1

Let's analyze each equation one by one to see which ones are true for the number 9.

1. **4 divide by ____ = 9/4**
To find the missing number, we set up the equation:
\[
\frac{4}{x} = \frac{9}{4}
\]
Cross-multiplying gives:
\[
4 \cdot 4 = 9 \cdot x \implies 16 = 9x \implies x = \frac{16}{9}
\]
This equation is true if \(x = \frac{16}{9}\).

2. **____ 1/9 = ____ divide by 9**
Let's set it up. Let \(a\) be the first blank and \(b\) be the second blank:
\[
\frac{a}{9} = \frac{b}{9}
\]
From this, we see that \(a = b\). However, we can't determine specific values for \(a\) and \(b\) just from the equation. But for \(a = b\), if we choose \(a = 1\) and \(b = 1\), the equation holds true. So, this equation can be true for certain values.

3. **____ 7 divide by _____ = 7/9**
Let’s denote the blanks as \(c\) and \(d\):
\[
\frac{c}{d} = \frac{7}{9}
\]
Rearranging gives:
\[
9c = 7d \implies c = \frac{7d}{9}
\]
This equation is true if \(c\) and \(d\) maintain the ratio of \(7:9\). For example, if \(d = 9\), then \(c = 7\).

4. **___ ___ divide by 1/9 = 81**
Denote the two blanks as \(e\) and \(f\):
\[
\frac{e}{1/9} = 81
\]
This can be rewritten as:
\[
e \cdot 9 = 81 \implies e = \frac{81}{9} = 9
\]
So, if \(e = 9\), then the equation holds true.

From this analysis, the following equations are true for the number 9:

- **4 divide by ____ = 9/4** (True if the blank is 16/9)
- **____ 1/9 = ____ divide by 9** (True if both blanks are equal)
- **____ 7 divide by _____ = 7/9** (True if the blanks maintain the ratio 7:9)
- **___ ___ divide by 1/9 = 81** (True if one blank is 9)

All equations have valid interpretations that can include the number 9. Thus, all four statements are potentially true based on the specified constraints.