Question

Consider the fractions \(\frac{2}{3}, \frac{5}{6}\) and \(\frac{7}{8}\). (a) Calculate the l.c.m. of their denominators. (b) Express each fraction as an equivalent fraction with the l.c.m. found in (a) as the denominator.

Short answer

Answer: The equivalent fractions with the least common multiple of the denominators are \(\frac{16}{24}, \frac{20}{24}\) and \(\frac{21}{24}\).

Step-by-step solution

List the prime factors of each denominator

First, we need to find the prime factors of each denominator (3, 6, and 8). Prime factors are numbers that are prime and, when multiplied, result in the original number. 3: 3 (since 3 is a prime number, its only prime factor is itself) 6: 2 x 3 8: 2 x 2 x 2

Determine the l.c.m. of the denominators

Now we need to find the least common multiple (l.c.m.) of the denominators. The l.c.m. is the smallest positive integer that is a multiple of all three denominators. l.c.m(3, 6, 8) = 2^3 x 3 = 8 x 3 = 24 So, the l.c.m. of the denominators is 24.

Express each fraction with a denominator of 24

Now that we have the l.c.m. of the denominators (24), we need to express each fraction as an equivalent fraction with 24 as the denominator. For \(\frac{2}{3}\), we have: \(\frac{2}{3} \times \frac{8}{8} = \frac{2 \times 8}{3 \times 8} = \frac{16}{24}\) For \(\frac{5}{6}\), we have: \(\frac{5}{6} \times \frac{4}{4} = \frac{5 \times 4}{6 \times 4} = \frac{20}{24}\) For \(\frac{7}{8}\), we have: \(\frac{7}{8} \times \frac{3}{3} = \frac{7 \times 3}{8 \times 3} = \frac{21}{24}\) So the equivalent fractions with l.c.m as denominator are \(\frac{16}{24}, \frac{20}{24}\) and \(\frac{21}{24}\).

Key concepts

Equivalent Fractions

When you work with fractions, understanding the concept of equivalent fractions is vital. Equivalent fractions are different fractions that represent the same part of a whole. Think of it like this: if you slice a pizza into four pieces and take two, you have half a pizza. Now, if instead you slice the pizza into eight pieces and take four, it might look different, but you still only have half a pizza.

In the case of our exercise, we are looking to express fractions as equivalent fractions with a new common denominator. To do this, we find a number that all the original denominators can be multiplied into, the least common multiple (LCM), and then adjust the numerators accordingly so the value of the fraction remains unchanged. For instance, \(\frac{2}{3}\) was turned into \(\frac{16}{24}\) by multiplying the numerator and the denominator by eight. This process ensures that we are comparing 'apples to apples' when we look at the fractions side by side.

Prime Factorization

Prime factorization is breaking down a number into its basic building blocks—the prime numbers that multiply together to give the original number. A prime number, remember, is a number greater than 1 that only has two factors: 1 and itself. This method is important because it reveals the 'DNA' of a number, and that can be extremely helpful in various mathematical operations, particularly when dealing with fractions.

In the example problem, we used prime factorization to break down the denominators. Three is already a prime number, so its factors are just itself. Six can be divided into the prime numbers two and three, and eight is two to the third power. These prime factors are then used to determine the least common multiple, which is essential in creating equivalent fractions with the same denominator.

LCM of Denominators

The least common multiple (LCM) of a set of denominators is the smallest number that each of the denominators can divide into without leaving a remainder. It's the shared multiple that's closest to zero above all those denominators. You can find the LCM by listing out multiples of each number until you find the smallest one they all share, but using prime factorization is often more efficient.

Once you obtain the prime factors of each denominator, you take the highest power of each prime that appears in the factorizations. In the given problem, the highest power of two from the denominators 3, 6, and 8 is three (from the number 8), and then we have a three from the number six. Multiplying these gets you the LCM of 24. The LCM is extremely useful when adding, subtracting, or comparing fractions because it allows us to rewrite them with a common denominator, which is a fundamental step in these operations.