My Homework Lesson 5 Simplest Form Answer Key

Working Through Simplest Form Problems

Most students hit a wall with Lesson 5 because they try to eyeball the greatest common factor instead of calculating it. That approach works fine for something like 4/8, but fall apart completely when you're staring at 48/72 or 90/126. The actual process is straightforward, even if the arithmetic gets tedious. First, identify the GCF of your numerator and denominator. You can use prime factorization, the ladder method, or just successive division. I personally switch between prime factorization for smaller numbers and the Euclidean algorithm for anything above two digits. The Euclidean algorithm is faster once you know it, though most textbooks don't teach it until later grades.

My Homework Lesson 5 Simplest Form Answer Key

When I was going through this lesson myself, I ran into a problem that wasn't in any of the standard answer keys. The fraction was 63/84, and the key said the answer was 3/4. I double-checked my work three times and kept getting 3/4 as well, but I couldn't figure out which GCF they had used. Turns out the textbook listed 21 as the GCF, but I had factored it differently and got there through a longer path. That's the thing nobody warns you about — there are multiple valid routes to the same simplified answer, and some answer keys only show one. If your final reduced fraction matches but your intermediate steps look different, you're not wrong. Let me walk through a full example. Take 48/72. The prime factors of 48 are 2 × 2 × 2 × 2 × 3, which is 2 × 3. The prime factors of 72 are 2 × 2 × 2 × 3 × 3, which is 2³ × 3². The common factors are 2³ × 3, equaling 24. Divide both by 24 and you get 2/3. That's the simplest form because 2 and 3 share no common factors other than 1. Another common one that trips people up: 90/126. Break 90 down to 2 × 3² × 5. Break 126 down to 2 × 3² × 7. The GCF here is 2 × 3² = 18. Divide top and bottom by 18 and you get 5/7. I've seen a lot of students divide by 6 instead and end up with 15/21, which looks simplified but isn't. They miss the second factor of 3 because they're rushing through the prime factorization step.

Here's the counter-intuitive part that beginners almost always miss. Sometimes the fraction looks like it needs heavy simplification but actually doesn't. For instance, 17/51 looks messy, but 17 divides evenly into 51 exactly three times, so the answer is 1/3. Students will sit there doing long prime factorizations on a prime numerator when the quick check is just to see if the denominator is a multiple of the numerator. If it is, the fraction reduces immediately. If it isn't, then you move to full prime factorization. That one check alone saves maybe 30 seconds per problem, and in a timed assignment it adds up. The other thing worth knowing is that some answer keys use the Euclidean algorithm and show a different step-by-step than what your teacher expects. The final answer is identical, but the method looks foreign if you've only been taught prime factorization. When your work and the answer key disagree on the method but agree on the result, submit your version and note the GCF you used. Teachers generally accept any valid approach. There are real limitations to this lesson format. The problems tend to cluster around three or four difficulty levels, which means you might breeze through six easy ones and then hit a wall on the last problem that requires factoring numbers in the hundreds. It's not a great design for building progressive confidence. Also, many of these answer keys online are scanned images of low resolution or contain typos in the denominators. I've personally encountered answer keys where 12/18 was marked as 3/2 instead of 2/3, which would send a confused student down a rabbit hole trying to figure out what they did wrong. Always verify by cross-multiplying — if your simplified fraction multiplied back gives you the original numerator and denominator, you're good.

When you can't find a reliable answer key, the best workaround is to simplify the fraction yourself using prime factorization, then check by dividing the original numerator by your simplified numerator to get the GCF, and confirming that the original denominator divided by that same GCF gives you your simplified denominator. It's a self-checking loop that catches most errors before you turn anything in. One more edge case from my own experience. A student once asked about 0/15. Technically the simplest form is 0, but some answer keys list it as 0/1 and that causes arguments over whether it's "fully simplified." The correct mathematical answer is just 0. If your key says 0/1, it's being overly pedantic. Push back on that one if it costs you points. For the actual problem sets, you're usually looking at fractions like 8/12, 15/25, 24/36, 35/49, and 40/55 as the easier set. The harder problems introduce larger numbers like 84/126, 100/140, and 72/96. Practice the Euclidean algorithm on those — it becomes significantly faster than prime factorization once you internalize the subtraction steps.