- Multiply the fraction by the whole number to find the required quantity, resulting in \(\frac{3}{5} \times 15 = \boxed{9}\) and \(\frac{2}{3} \times 30 = \boxed{20}\).
- Verify equivalent fractions by multiplying both the numerator and denominator by the same common factor to confirm they represent the same value: \(\frac{1}{2} = \frac{3}{6}\), \(\frac{3}{4} = \frac{6}{8}\), \(\frac{1}{3} = \frac{3}{9}\), \(\frac{2}{5} = \frac{4}{10}\), and \(\frac{2}{4} = \frac{4}{8}\).
- All provided equivalent fractions are correct, and the missing values are identified.
- Final answers: 3b = \(\boxed{9}\), 3c = \(\boxed{20}\).
Explanation
- Analysis of the Problem
We are tasked with solving a set of arithmetic problems involving fractions. Specifically, we need to calculate the values of "fractions of a whole" for given problems and verify the provided equivalent fractions.
- Problem 3b: Calculate \(\frac{3}{5}\) of \(15\).
- Problem 3c: Calculate \(\frac{2}{3}\) of \(30\).
- Problem 4a-e: Verify the equivalence of the given fractions.
These problems help solidify the understanding of fractional multiplication and the concept of equivalent fractions, where the ratio of the numerator to the denominator remains constant.
- Calculating Fractions of a Whole
To calculate the fraction of a whole number, we multiply the fraction by that number.
For 3b: \(\frac{3}{5}\) of \(15\) means \(\frac{3}{5} \times 15\). Dividing \(15\) by the denominator \(5\) gives \(3\), and multiplying that by the numerator \(3\) gives \(3 \times 3 = 9\).
For 3c: \(\frac{2}{3}\) of \(30\) means \(\frac{2}{3} \times 30\). Dividing \(30\) by the denominator \(3\) gives \(10\), and multiplying that by the numerator \(2\) gives \(2 \times 10 = 20\).
The results are: 3b = 9, 3c = 20.
- Verifying Equivalent Fractions
Equivalent fractions represent the same portion of a whole. A fraction \(\frac{a}{b}\) is equivalent to \(\frac{c}{d}\) if multiplying or dividing both \(a\) and \(b\) by the same non-zero number results in \(c\) and \(d\).
- 4a: \(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\). Verified.
- 4b: \(\frac{3}{4} = \frac{3 \times 2}{4 \times 2} = \frac{6}{8}\). Verified.
- 4c: \(\frac{1}{3} = \frac{1 \times 3}{3 \times 3} = \frac{3}{9}\). Verified.
- 4d: \(\frac{2}{5} = \frac{2 \times 2}{5 \times 2} = \frac{4}{10}\). Verified.
- 4e: \(\frac{2}{4} = \frac{2 \times 2}{4 \times 2} = \frac{4}{8}\). Verified.
All given equivalent fractions are correct.
- Final Results
The calculations for the requested fractions of a whole and the verification of the equivalent fractions have been completed.
The results are:
- 3b: \(\frac{3}{5}\) of \(15 = \boxed{9}\)
- 3c: \(\frac{2}{3}\) of \(30 = \boxed{20}\)
- 4a: \(\frac{1}{2} = \frac{3}{6}\)
- 4b: \(\frac{3}{4} = \frac{6}{8}\)
- 4c: \(\frac{1}{3} = \frac{3}{9}\)
- 4d: \(\frac{2}{5} = \frac{4}{10}\)
- 4e: \(\frac{2}{4} = \frac{4}{8}\)
Examples
Imagine you are baking cookies and the recipe requires \(\frac{3}{5}\) of a 15-ounce bag of chocolate chips. To know how many ounces to use, you calculate \(\frac{3}{5} \times 15 = 9\) ounces. Equivalent fractions help when you only have a measuring cup of a different size; if you need \(\frac{1}{2}\) cup but your smallest cup is \(\frac{1}{6}\), knowing \(\frac{1}{2} = \frac{3}{6}\) tells you to fill that cup 3 times! Mastering these concepts makes kitchen measurements much simpler and more accurate.