Solutions

1. \(\frac{4}{6}=\frac{2\times2}{3\times2}=\frac{2}{3}\) (Written Solution)

   This fraction bar represents 4 out of 6 or 4/6. Four are shaded orange out of a total of six.

   

   To simplify the fraction, divide the numerator and denominator by 2.

   

   To simplify the fraction bar, group 2 rectangles into one bar. This is the same as to divide by 2.

   
   

   The resulting fraction is \(\frac{2}{3}\); and 2 out of 3 bars are shaded orange.

   



2. \(\frac{10}{25}=\frac{2\times5}{5\times5}=\frac{2}{5}\) (Written Solution)

   We can simplify a fraction if the number in the numerator and the number in the denominator share at least one common factor.

   We must first find the factorizations of these two numbers.

   • Prime factorization of 10:

   10 is divisible by 2 since it is an even number.

   2 × 5 = 10

   

   Both the number 2 and the number 5 are prime numbers so the prime factorization of 10 is 2 × 5

   • Prime factorization of 25:

   25 ends in a 5 and any number that ends in a 5 or a 0 is divisible by 5.

   5 × 5 = 25

   

   The number 5 is a prime number so the prime factorization of 25 is simply 5 × 5.

   Now that we have the prime factorization of both the numerator and the denominator, we can rewrite the fraction like this: \(\frac{10}{25}=\frac{2\times5}{5\times5}\)

   

   Note: A dot between numbers represents multiplication. This dot is centered between the numbers and higher than a decimal point.

   Since the numerator and the denominator both share one 5, and since \(\frac{5}{5}=1\) we can rewrite the fraction as \(\frac{2\times5}{5\times5}=\frac{2}{5}\times1\)

   

   Anything multiplied by 1 is just itself, so \(\frac{2}{5}\times1=\frac{2}{5}\).

   There are no other common factors between 2 and 5 so this fraction is as simplified as it can be.

   Our final solution: \(\frac{2}{5}\)



3. \(\frac{4}{7}\) is already simplified to lowest terms. (Written Solution)

   The 4 and 7 do not share a common factor other than 1, therefore this answer is written in its simplest or lowest terms.



4. \(\frac{30}{48}=\frac{2\times3\times5}{2\times2\times2\times2\times3}=\frac{5}{8}\) (Solution Video | Transcript)


5. \(\frac{42}{70}=\frac{2\times3\times7}{2\times7\times7}=\frac{3}{5}\) (Solution Video | Transcript)


6. \(\frac{12}{84}=\frac{2\times2\times3}{2\times2\times3\times7}=\frac{1}{7}\) (Written Solution)

   If the numerator and denominator of a fraction have any common factors, the fraction can be simplified.

   • First, we will find the prime factorization of the numerator:

   12 is divisible by 2 because it is an even number and all even numbers are divisible by 2.

   2 × 6 = 12

   

   6 is not prime so now we must factor 6 as well.

   6 is also even so we can divide it by 2 as well.

   2 × 3 = 6

   The numbers 2 and 3 are both prime, so this is as far as we can factor our number.

   

   The prime factorization of 12 is 2 × 2 × 3.

   • Next, we will find the prime factorization of the denominator:

   84 is even so we will start by dividing it by 2.

   2 × 42 = 84

   

   The number 2 is prime but 42 is not so we still need to find the factors of 42.

   42 is also even so we can divide it by 2 as well.

   2 × 21 = 42

   

   The number 2 is prime but 21 is not prime, so we need to factor 21 as well.

   21 is divisible by 3.

   3 × 7 = 21

   

   Both 3 and 7 are prime, so this is as far as we can factor.

   The prime factorization of 21 is 2 × 2 × 3 × 7.

   Now we can use the prime factorizations to determine if there are any common factors in the numerator and the denominator.

   Note: A dot between numbers represents multiplication. This dot is centered between the numbers and higher than a decimal point.

   \(\frac{12}{84}=\frac{2\cdot2\cdot3}{2\cdot2\cdot3\cdot7}\) Here we see that both the numerator and the denominator have two 2s and a 3 in common. We can rewrite the fraction like this:

   \(\frac{12}{84}=\frac{2\times2\times3}{2\times2\times3\times7}=\frac{2}{2}\times\frac{2}{2}\times\frac{3}{3}\times\frac{1}{7}\)

   Note: you might think that there is only 0 left in the numerator after separating out the 2 × 2 × 3, but really, there is still a 1 because if there was a 0 in the numerator, the numerator would equal 0. There is always an invisible 1 being multiplied to everything. 2 × 2 × 3 × 1 = 12.

   Anything divided by itself is equal to 1.

   \(\frac{2}{2}\cdot\frac{2}{2}\cdot\frac{3}{3}\cdot\frac{1}{7}=1\cdot1\cdot1\cdot\frac{1}{7}\)

   Anything multiplied by 1 is just itself, so we are just left with \(\frac{1}{7}\).

   \(\frac{12}{84}\) simplifies to \(\frac{1}{7}\).