Sure! Let's solve the given equation step-by-step:
### Given equation:
[tex]\[ 1 \frac{2}{3} \div 5.5 + 1 \frac{7}{12} \div (2.25 - \square) = 1 \frac{1}{6} \][/tex]
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert all mixed numbers to improper fractions for easier calculation.
- [tex]\( 1 \frac{2}{3} = \frac{5}{3} \)[/tex]
- [tex]\( 1 \frac{7}{12} = \frac{19}{12} \)[/tex]
- [tex]\( 1 \frac{1}{6} = \frac{7}{6} \)[/tex]
### Step 2: Rewrite the Equation with Improper Fractions and Decimals
Let's rewrite the equation using these improper fractions:
[tex]\[ \frac{5}{3} \div 5.5 + \frac{19}{12} \div (2.25 - \square) = \frac{7}{6} \][/tex]
### Step 3: Convert Division to Multiplication by Reciprocal
We know that dividing by a number is the same as multiplying by its reciprocal. So:
[tex]\[ \frac{5}{3} \times \frac{1}{5.5} + \frac{19}{12} \times \frac{1}{(2.25 - \square)} = \frac{7}{6} \][/tex]
### Step 4: Simplify the First Term
Next, simplify the first term on the left side:
[tex]\[ \frac{5}{3} \times \frac{1}{5.5} = \frac{5}{3} \times \frac{1}{\frac{11}{2}} = \frac{5}{3} \times \frac{2}{11} = \frac{10}{33} \][/tex]
### Step 5: Rewrite the Equation
Now, the equation looks like this:
[tex]\[ \frac{10}{33} + \frac{19}{12} \times \frac{1}{(2.25 - \square)} = \frac{7}{6} \][/tex]
Let's denote [tex]\( \square \)[/tex] as [tex]\( x \)[/tex] and substitute it.
[tex]\[ \frac{10}{33} + \frac{19}{12} \times \frac{1}{(2.25 - x)} = \frac{7}{6} \][/tex]
### Step 6: Isolate the Fraction with [tex]\( x \)[/tex]
Subtract [tex]\(\frac{10}{33}\)[/tex] from both sides:
[tex]\[ \frac{19}{12} \times \frac{1}{(2.25 - x)} = \frac{7}{6} - \frac{10}{33} \][/tex]
### Step 7: Find the Common Denominator and Simplify
The common denominator of 6 and 33 is 66:
[tex]\[ \frac{7}{6} = \frac{77}{66} \][/tex]
[tex]\[ \frac{10}{33} = \frac{20}{66} \][/tex]
Subtract these fractions:
[tex]\[ \frac{77}{66} - \frac{20}{66} = \frac{57}{66} = \frac{19}{22} \][/tex]
So now the equation is:
[tex]\[ \frac{19}{12} \times \frac{1}{(2.25 - x)} = \frac{19}{22} \][/tex]
### Step 8: Eliminate the Fraction by Multiplying Both Sides
Multiply both sides by [tex]\( \frac{12}{19} \)[/tex]:
[tex]\[ \frac{1}{(2.25 - x)} = \frac{12}{22} = \frac{6}{11} \][/tex]
### Step 9: Solve for [tex]\( x \)[/tex]
Take the reciprocal of both sides:
[tex]\[ 2.25 - x = \frac{11}{6} \][/tex]
Subtract [tex]\(\frac{11}{6}\)[/tex] from [tex]\(2.25\)[/tex]:
Convert [tex]\(2.25\)[/tex] to a fraction:
[tex]\[ 2.25 = \frac{9}{4} \][/tex]
Now, find the common denominator, which is 12:
[tex]\[ \frac{9}{4} = \frac{27}{12} \][/tex]
[tex]\[ \frac{11}{6} = \frac{22}{12} \][/tex]
Subtract these fractions:
[tex]\[ \frac{27}{12} - \frac{22}{12} = \frac{5}{12} \][/tex]
So, [tex]\( x = \frac{5}{12} \)[/tex].
### Final Answer:
[tex]\[ \boxed{\frac{5}{12}} \][/tex]
