Answer :
To solve the given numerical expression and identify where the first mistake occurs, let's evaluate the expression step by step:
The expression given is:
[tex]\[ 2 \frac{1}{2} + \frac{1}{3} \times 3 \frac{3}{4} \times 7 - 21 \][/tex]
We need to evaluate the expression by following the order of operations, which is commonly remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
1. Convert Mixed Numbers to Improper Fractions:
- [tex]\( 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{5}{2} \)[/tex]
- [tex]\( 3 \frac{3}{4} = 3 + \frac{3}{4} = \frac{15}{4} \)[/tex]
2. Evaluate the Multiplication:
First, we perform the multiplication inside the expression:
[tex]\[
\frac{1}{3} \times \frac{15}{4} \times 7
\][/tex]
Step by step:
- [tex]\( \frac{1}{3} \times \frac{15}{4} = \frac{15}{12} \)[/tex]
- [tex]\( \frac{15}{12} \times 7 = \frac{105}{12} \)[/tex]
3. Add and Subtract:
Now, insert this result into the expression:
[tex]\[
\frac{5}{2} + \frac{105}{12} - 21
\][/tex]
First, we need to add or identify missteps:
- Convert [tex]\(\frac{5}{2}\)[/tex] to a common denominator with [tex]\(\frac{105}{12}\)[/tex], which is 12:
[tex]\(\frac{5}{2} = \frac{30}{12}\)[/tex]
Now, add them:
[tex]\[
\frac{30}{12} + \frac{105}{12} = \frac{135}{12}
\][/tex]
Then, subtract 21:
- First, express 21 with the same denominator:
[tex]\( 21 = \frac{252}{12} \)[/tex]
[tex]\[
\frac{135}{12} - \frac{252}{12} = \frac{-117}{12}
\][/tex]
[tex]\(\frac{-117}{12}\)[/tex] simplifies to approximately [tex]\(-9.75\)[/tex].
Let's review the steps for any mistakes:
- Step 1 (`A`):
Perhaps [tex]\(2 \frac{1}{2} = \frac{5}{2}\)[/tex] should have been evaluated appropriately in fractional format or another operation included in steps.
- Step 2 (`B`): Multiplies the parts but [tex]\(\frac{15}{12} \times 7\)[/tex] is [tex]\(\frac{105}{12}\)[/tex], not [tex]\(\frac{45}{12} \times 7\)[/tex].
Upon evaluating, the first mistake appears in Step 2 (`B`), where incorrect multiplication results are used. The evaluation should be carried out through precise steps, verifying transformations—so finding where misalignments or assumptions occur is essential.
The expression given is:
[tex]\[ 2 \frac{1}{2} + \frac{1}{3} \times 3 \frac{3}{4} \times 7 - 21 \][/tex]
We need to evaluate the expression by following the order of operations, which is commonly remembered by PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)).
1. Convert Mixed Numbers to Improper Fractions:
- [tex]\( 2 \frac{1}{2} = 2 + \frac{1}{2} = \frac{5}{2} \)[/tex]
- [tex]\( 3 \frac{3}{4} = 3 + \frac{3}{4} = \frac{15}{4} \)[/tex]
2. Evaluate the Multiplication:
First, we perform the multiplication inside the expression:
[tex]\[
\frac{1}{3} \times \frac{15}{4} \times 7
\][/tex]
Step by step:
- [tex]\( \frac{1}{3} \times \frac{15}{4} = \frac{15}{12} \)[/tex]
- [tex]\( \frac{15}{12} \times 7 = \frac{105}{12} \)[/tex]
3. Add and Subtract:
Now, insert this result into the expression:
[tex]\[
\frac{5}{2} + \frac{105}{12} - 21
\][/tex]
First, we need to add or identify missteps:
- Convert [tex]\(\frac{5}{2}\)[/tex] to a common denominator with [tex]\(\frac{105}{12}\)[/tex], which is 12:
[tex]\(\frac{5}{2} = \frac{30}{12}\)[/tex]
Now, add them:
[tex]\[
\frac{30}{12} + \frac{105}{12} = \frac{135}{12}
\][/tex]
Then, subtract 21:
- First, express 21 with the same denominator:
[tex]\( 21 = \frac{252}{12} \)[/tex]
[tex]\[
\frac{135}{12} - \frac{252}{12} = \frac{-117}{12}
\][/tex]
[tex]\(\frac{-117}{12}\)[/tex] simplifies to approximately [tex]\(-9.75\)[/tex].
Let's review the steps for any mistakes:
- Step 1 (`A`):
Perhaps [tex]\(2 \frac{1}{2} = \frac{5}{2}\)[/tex] should have been evaluated appropriately in fractional format or another operation included in steps.
- Step 2 (`B`): Multiplies the parts but [tex]\(\frac{15}{12} \times 7\)[/tex] is [tex]\(\frac{105}{12}\)[/tex], not [tex]\(\frac{45}{12} \times 7\)[/tex].
Upon evaluating, the first mistake appears in Step 2 (`B`), where incorrect multiplication results are used. The evaluation should be carried out through precise steps, verifying transformations—so finding where misalignments or assumptions occur is essential.
