Verified Answer :

  • Identify the Least Common Multiple (LCM) of 35 and 15, which is 105.
  • Convert both fractions to have 105 as their denominator: \(-\frac{3}{105}\) and \(-\frac{7}{105}\).
  • Subtract the numerators: \(-3 - 7 = -10\), resulting in \(-\frac{10}{105}\).
  • Simplify the fraction by dividing the numerator and denominator by 5 to obtain \(\boxed{-\frac{2}{21}}\).

Explanation

  1. Finding the Common Denominator

We need to evaluate the expression \(-\frac{1}{35} - \frac{1}{15}\). To perform subtraction of these two fractions, they must have a common denominator. The denominators are 35 and 15. We start by finding the prime factorization of each denominator to determine their Least Common Multiple (LCM):

  • \(35 = 5 \times 7\)
  • \(15 = 3 \times 5\)

The LCM is the product of the highest power of each prime factor, which is \(3 \times 5 \times 7 = 105\).

  1. Performing the Subtraction

Next, we convert each fraction to an equivalent fraction with the denominator 105:

  • For the first fraction: \(-\frac{1}{35} = -\frac{1 \times 3}{35 \times 3} = -\frac{3}{105}\)
  • For the second fraction: \(-\frac{1}{15} = -\frac{1 \times 7}{15 \times 7} = -\frac{7}{105}\)

Now, combine them:
\(-\frac{3}{105} - \frac{7}{105} = \frac{-3 - 7}{105} = -\frac{10}{105}\)

  1. Simplifying the Result

Finally, we simplify the fraction \(-\frac{10}{105}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 5:

  • Numerator: \(-10 \div 5 = -2\)
  • Denominator: \(105 \div 5 = 21\)

Thus, the expression simplifies to \(-\frac{2}{21}\).

Examples

When cooking, you might find yourself needing to subtract fractional portions of ingredients. For example, if you are scaling down a recipe and need to remove \(\frac{1}{35}\) of a kilogram of flour and then another \(\frac{1}{15}\) of a kilogram because of a change in your plans, you are performing fraction arithmetic. Mastering these operations ensures you maintain the correct proportions for your dishes, just as we simplified the expression here to get a precise, usable result.

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