We're going to learn how to simplify fractions. Let's start with the fraction 30 over 42. Fractions can be written in different forms that represent the same value. Simplifying makes them easier to work with and understand.
To simplify a fraction, we need to find the greatest common factor of the numerator and denominator. Let's find all the factors of 30 and 42. For 30, we have 1 times 30, 2 times 15, 3 times 10, and 5 times 6. For 42, we have 1 times 42, 2 times 21, 3 times 14, and 6 times 7. The common factors are 1, 2, 3, and 6.
Now we need to identify the greatest common factor, which is the largest number that divides both the numerator and denominator. We can use the prime factorization method. Thirty equals two times three times five. Forty-two equals two times three times seven. The common prime factors are two and three. Multiplying them together gives us the GCF of six.
Now we apply the GCF to simplify the fraction. We divide both the numerator and denominator by six. Thirty divided by six equals five, and forty-two divided by six equals seven. So thirty-forty-seconds simplifies to five-sevenths. This division maintains the fraction's value while making it simpler.
Let's verify our answer. We have the original fraction thirty-forty-seconds and the simplified fraction five-sevenths. To confirm they're equivalent, we use cross-multiplication. Thirty times seven equals two hundred ten, and forty-two times five also equals two hundred ten. Since these products are equal, the fractions are equivalent. Five and seven are both prime numbers with no common factors other than one, confirming that five-sevenths is in lowest terms. Therefore, thirty-forty-seconds simplified equals five-sevenths.