Simplify the following:
(i) $\left[3 \frac{1}{9} \times 2 \frac{4}{7}\right] \times 1 \frac{7}{8}$
Answer (2)
The simplified value of the expression [ 3 9 1 × 2 7 4 ] × 1 8 7 is 15 . This is achieved by converting mixed numbers to improper fractions and performing multiplication step-by-step. The final answer is 15 .
;
Convert the mixed fractions to improper fractions: 3 9 1 = 9 28 , 2 7 4 = 7 18 , and 1 8 7 = 8 15 .
Multiply the first two fractions: 9 28 × 7 18 = 8 .
Multiply the result by the remaining fraction: 8 × 8 15 = 15 .
The simplified expression is 15 .
Explanation
Understanding the problem We are asked to simplify the expression [ 3 r a c 1 9 × 2 r a c 4 7 ] × 1 r a c 7 8 . To do this, we will first convert the mixed fractions to improper fractions, then perform the multiplication, and finally simplify the result.
Converting to improper fractions First, convert the mixed fractions to improper fractions:
3 r a c 1 9 = r a c 3 × 9 + 1 9 = r a c 27 + 1 9 = r a c 28 9
2 r a c 4 7 = r a c 2 × 7 + 4 7 = r a c 14 + 4 7 = r a c 18 7
1 r a c 7 8 = r a c 1 × 8 + 7 8 = r a c 8 + 7 8 = r a c 15 8
Substituting improper fractions Now, substitute these improper fractions into the original expression:
[ \f r a c 28 9 × \f r a c 18 7 ] × \f r a c 15 8
Multiplying fractions inside the brackets Next, perform the multiplication inside the brackets:
\f r a c 28 9 × \f r a c 18 7 = r a c 28 × 18 9 × 7
Simplifying the fraction Simplify the fraction by canceling common factors:
\f r a c 28 × 18 9 × 7 = r a c 4 × 7 × 2 × 9 9 × 7 = 4 × 2 = 8
Multiplying by the remaining fraction Now, multiply the result by \f r a c 15 8 :
8 × \f r a c 15 8 = r a c 8 × 15 8
Simplifying the final expression Finally, simplify the expression by canceling the common factor of 8:
\f r a c 8 × 15 8 = 15
Final Answer Therefore, the simplified expression is 15.
Examples
Understanding how to simplify expressions with mixed fractions is useful in everyday situations, such as when you're cooking and need to adjust recipe quantities. For example, if a recipe calls for 2 2 1 cups of flour and you want to triple the recipe, you'll need to multiply 2 2 1 by 3. Converting this to improper fractions makes the calculation easier: 2 5 × 3 = 2 15 = 7 2 1 cups of flour.
