Bro and Sis Math Club: Estimating the Sums and Differences of ... - Free Printable

Let’s solve each problem one by one. We’ll simplify each fraction first, then add or subtract, and finally reduce to the least form.

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Problem 1:
6/36 + 3/24

Step 1: Simplify each fraction.
- 6/36 = 1/6 (divide numerator and denominator by 6)
- 3/24 = 1/8 (divide numerator and denominator by 3)

Now we have: 1/6 + 1/8

Step 2: Find a common denominator. LCM of 6 and 8 is 24.
- 1/6 = 4/24
- 1/8 = 3/24

Add: 4/24 + 3/24 = 7/24

7 and 24 have no common factors → already in least form.

✔ Answer: 7/24

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Problem 2:
33/33 + 22/22

Step 1: Simplify each.
- 33/33 = 1
- 22/22 = 1

Add: 1 + 1 = 2

✔ Answer: 2

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Problem 3:
4/13 + 6/26

Step 1: Simplify 6/26 → divide by 2 → 3/13

Now: 4/13 + 3/13 = 7/13

Already simplified.

✔ Answer: 7/13

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Problem 4:
7/9 + 6/7

Step 1: No simplification possible for either fraction.

Find common denominator: LCM of 9 and 7 is 63.

Convert:
- 7/9 = (7×7)/(9×7) = 49/63
- 6/7 = (6×9)/(7×9) = 54/63

Add: 49/63 + 54/63 = 103/63

Check if it can be simplified: 103 is prime, doesn’t divide 63 → leave as improper fraction or convert to mixed number?

The instruction says “simplify to the least form” — usually means reduced fraction, not necessarily mixed number unless specified. But 103/63 is already reduced.

But let’s check: 103 ÷ 63 = 1 with remainder 40 → so 1 40/63. Since 40 and 63 share no common factors, this is also acceptable.

However, since the original problems are all proper fractions except #2, and the instruction says “least form”, which typically means lowest terms, not necessarily mixed number — but sometimes “least form” implies mixed number if improper. Let’s see what’s standard.

In most school contexts, “simplest form” for an improper fraction like 103/63 is left as is unless asked to write as mixed number. But to be safe, let’s note both.

Actually, looking at other problems — none result in improper fractions except maybe this one. Let’s double-check calculation:

7/9 + 6/7 = (49 + 54)/63 = 103/63 → correct.

103 and 63: GCD? 63 = 7×9, 103 not divisible by 7 or 3 → yes, 103/63 is simplest.

But perhaps they expect mixed number? The problem says “simplify to the least form” — in many curricula, that includes converting improper fractions to mixed numbers.

Let me check Problem 5 and others later — but for now, I’ll go with 103/63, but I think better to write as mixed number.

Wait — let’s look at Problem 5: 9/9 + 5/5 = 1+1=2 — integer.

Problem 7: 23/46 - 24/48 = 1/2 - 1/2 = 0 — fine.

So for consistency, if we get an improper fraction, we should convert to mixed number only if required. But “least form” usually means reduced fraction, regardless of being improper.

To avoid confusion, I’ll keep it as 103/63, but let me verify with another approach.

Alternatively, maybe I made a mistake? 7/9 ≈ 0.777, 6/7≈0.857, sum≈1.634, and 103/63≈1.634 — correct.

I think 103/63 is acceptable, but let’s see what the expected answer might be. Since the problem says “simplify to the least form”, and 103/63 cannot be simplified further, it’s fine.

But actually, in many textbooks, they prefer mixed numbers for final answers when improper. Let me check online or recall — for example, Khan Academy often accepts either, but prefers mixed numbers for clarity.

Given that, and since this is for a student, I’ll convert to mixed number.

103 ÷ 63 = 1, remainder 40 → so 1 40/63

And 40 and 63 have no common factors (GCD=1), so 1 40/63 is simplest mixed form.

I think that’s better.

✔ Answer: 1 40/63

But wait — the original problems don’t have mixed numbers in input, so output might be expected as improper. Hmm.

Looking back at the worksheet title: “Estimating the Sums and Differences of Fractions” — but the instruction is “Simplify to the least form”, which suggests exact calculation, not estimation.

Perhaps I should stick to reduced fraction.

To resolve this, let’s look at Problem 5: 9/9 + 5/5 = 1 + 1 = 2 — integer.

Problem 8: 15/135 - 2/5 = 1/9 - 2/5 = negative, etc.

For Problem 4, I’ll provide 103/63, but I suspect the expected answer is 1 40/63.

Let me calculate numerically: 7/9 + 6/7 = (49 + 54)/63 = 103/63 ≈ 1.6349

1 40/63 = 103/63 same thing.

I think for consistency with "least form", and since other answers are fractions, I'll leave as 103/63, but I recall that in some systems, "simplest form" for improper fractions is to write as mixed number.

Upon second thought, let's check the definition: "A fraction is in simplest form when the numerator and denominator have no common factors other than 1." It doesn't specify proper or improper. So 103/63 is in simplest form.

Moreover, in higher grades, improper fractions are preferred.

So I'll go with 103/63.

But let's see Problem 5: 9/9 + 5/5 = 1 + 1 = 2, which is integer.

For Problem 4, 103/63 is fine.

I'll proceed.

✔ Final for 4: 103/63

But I'm unsure. Let me skip and come back.

Actually, let's do all and see.

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Problem 5:
9/9 + 5/5

Simplify:
- 9/9 = 1
- 5/5 = 1

1 + 1 = 2

✔ Answer: 2

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Problem 6:
7/63 - 1/3

Simplify 7/63: divide by 7 → 1/9

So: 1/9 - 1/3

Common denominator: 9

1/3 = 3/9

So: 1/9 - 3/9 = -2/9

Negative is fine.

✔ Answer: -2/9

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Problem 7:
23/46 - 24/48

Simplify:
- 23/46 = 1/2 (divide by 23)
- 24/48 = 1/2 (divide by 24)

So: 1/2 - 1/2 = 0

✔ Answer: 0

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Problem 8:
15/135 - 2/5

Simplify 15/135: divide by 15 → 1/9

So: 1/9 - 2/5

Common denominator: 45

1/9 = 5/45

2/5 = 18/45

So: 5/45 - 18/45 = -13/45

-13 and 45 have no common factors.

✔ Answer: -13/45

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Problem 9:
13/63 - 7/14

Simplify 7/14 = 1/2

So: 13/63 - 1/2

Common denominator: LCM of 63 and 2 is 126

13/63 = (13×2)/(63×2) = 26/126

1/2 = 63/126

So: 26/126 - 63/126 = -37/126

Check if reducible: 37 is prime, 126 ÷ 37? No, so -37/126 is simplest.

✔ Answer: -37/126

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Now back to Problem 4: 7/9 + 6/7 = 103/63

Is 103/63 reducible? 103 divided by primes: 2,3,5,7,11 — 103÷7≈14.7, not integer; 103÷3≈34.3, no; so yes, irreducible.

But should it be written as mixed number? In the context of "simplify to the least form", and since other answers include integers and negatives, I think 103/63 is acceptable. However, to match typical elementary/middle school expectations, they might want mixed number.

Let me see the worksheet — it has fractions like 6/36, which simplify to 1/6, so probably expects reduced fractions, and for improper, perhaps mixed.

I recall that in many such worksheets, if the result is improper, they ask to write as mixed number.

For example, in Problem 2, they got 2, which is integer.

For Problem 4, 103/63 = 1 40/63, and 40/63 is reduced.

I think it's safer to write as mixed number for clarity.

Moreover, the instruction "simplify to the least form" might imply making it as simple as possible, and mixed number is simpler to understand.

So I'll change to 1 40/63.

Similarly, for consistency.

So for Problem 4: 1 40/63

Now, let's list all answers:

1. 7/24
2. 2
3. 7/13
4. 1 40/63
5. 2
6. -2/9
7. 0
8. -13/45
9. -37/126

Double-check Problem 3: 4/13 + 6/26 = 4/13 + 3/13 = 7/13 — correct.

Problem 6: 7/63 = 1/9, 1/9 - 1/3 = 1/9 - 3/9 = -2/9 — correct.

Problem 9: 13/63 - 7/14 = 13/63 - 1/2 = 26/126 - 63/126 = -37/126 — correct.

All seem good.

Final Answers:

1. \frac{7}{24}
2. 2
3. \frac{7}{13}
4. 1\frac{40}{63}
5. 2
6. -\frac{2}{9}
7. 0
8. -\frac{13}{45}
9. -\frac{37}{126}

For Problem 4, I'll write it as a mixed number.

In the boxed answer, I need to present clearly.

Since the user might expect fractions, but mixed number is fine.

I think it's correct.

So Final Answer section will list them.