Answer:
So the final equation becomes:
[tex]{\frac{9}{10} \times \left( \frac{8}{9} + \frac{7}{8} \right) = \left( \frac{1143}{1288} \times \frac{8}{9} \right) + \left( \frac{9}{10} \times \frac{1143}{1288} \right).}[/tex]
Step-by-step explanation:
To solve the equation
[tex]{\frac{9}{10} \times \left( \frac{8}{9} + \frac{7}{8} \right) = ( \_ \times \frac{8}{9} ) + \left( \frac{9}{10} \times \_ \right)}[/tex]
we will first simplify the left-hand side.Step 1: Simplify the Left-Hand Side
1. Calculate
[tex]\frac{8}{9} + \frac{7}{8}[/tex]
- Find a common denominator, which is 72 :
[tex] \frac{8}{9} = \frac{8 \times 8}{9 \times 8} = \frac{64}{72}[/tex]
[tex] \frac{7}{8} = \frac{7 \times 9}{8 \times 9} = \frac{63}{72}[/tex]
Now add the two fractions:
[tex] \frac{8}{9} + \frac{7}{8} = \frac{64}{72} + \frac{63}{72} = \frac{127}{72}[/tex]
2. Now multiply by \frac{9}{10} :
[tex] \frac{9}{10} \times \frac{127}{72} = \frac{9 \times 127}{10 \times 72} = \frac{1143}{720}[/tex]
Step 2: Set Up the Right-Hand Side
Now we have:
[tex]\frac{1143}{720} = ( \_ \times \frac{8}{9} ) + \left( \frac{9}{10} \times \_ \right)[/tex]
Let x be the blank we need to fill in:
[tex]\frac{1143}{720} = (x \times \frac{8}{9}) + \left( \frac{9}{10} \times x \right)[/tex]
Step 3: Combine the Right-Hand Side
1. Factor out x :
[tex] \frac{1143}{720} = x \left( \frac{8}{9} + \frac{9}{10} \right)[/tex]
2. Find a common denominator for :
[tex] \frac{8}{9} + \frac{9}{10} [/tex]
- The common denominator is 90 :
[tex] \frac{8}{9} = \frac{8 \times 10}{9 \times 10} = \frac{80}{90}[/tex]
[tex] \frac{9}{10} = \frac{9 \times 9}{10 \times 9} = \frac{81}{90}[/tex]
- Now add the two fractions:
[tex]\frac{8}{9} + \frac{9}{10} = \frac{80}{90} + \frac{81}{90} = \frac{161}{90}[/tex]
Step 4: Substitute Back
Now we have:
[tex]\frac{1143}{720} = x \left( \frac{161}{90} \right).[/tex]
Step 5: Solve for x
1. Multiply both sides by:
[tex]\frac{90}{161}[/tex]
[tex]x = \frac{1143}{720} \times \frac{90}{161}.[/tex]
2. Calculate x :
- First, simplify
[tex] \frac{1143 \times 90}{720 \times 161} [/tex]
[tex]x = \frac{1143 \times 90}{720 \times 161}[/tex]
3. Calculate 1143 x 90 :
[tex] 1143 \times 90 = 102870.[/tex]
4. Calculate 720 x 161 :
[tex]720 \times 161 = 115920.[/tex]
5. Now simplify:
[tex] x = \frac{102870}{115920}.[/tex]
6. Simplifying
[tex]\frac{102870}{115920} [/tex]
- Both numbers can be divided by 90 :
[tex]x = \frac{1143}{1288}.[/tex]
### Final Result
Thus, the values to fill in the blanks are:
[tex]\_ = \frac{1143}{1288}.[/tex]
