Proportions Worksheets - TheWorksheets.CoM - TheWorksheets Library - Free Printable

Let’s solve each problem step by step.

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Problem 1:
Solve: \( \frac{3}{4} + \frac{2}{5} \)

Step 1: Find a common denominator.
The least common denominator of 4 and 5 is 20.

Step 2: Convert fractions:
\( \frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20} \)
\( \frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20} \)

Step 3: Add them:
\( \frac{15}{20} + \frac{8}{20} = \frac{23}{20} \)

Answer: \( \frac{23}{20} \) or \( 1\frac{3}{20} \)

But the image shows “= 6” — that’s incorrect. Let’s check if it was meant to be something else? Wait — maybe it's not addition? The image says “+”, so we stick with addition. But 6 is wrong. Perhaps typo in image? We’ll go with correct math.

Wait — looking again at the image layout, maybe Problem 1 is actually:

It says:
“1) \( \frac{3}{4} + \frac{2}{5} \)” → then below it says “= 6” — but that’s clearly wrong. Maybe it’s a different operation? Or perhaps it’s multiplication?

Wait — let me re-express based on what’s written.

Actually, looking carefully (even though I can’t describe the image), from your instruction, you want me to solve accurately. So I will assume the problems are as written numerically.

But since you said “do not describe the image”, I must infer from standard interpretation.

Alternatively — perhaps the first one is:

Looking at typical homework sheets, sometimes they write:

1) \( \frac{3}{4} + \frac{2}{5} \) = ?

We did that: \( \frac{23}{20} \)

But the red text says “= 6” — which is likely an error in the original sheet. We ignore red answers unless instructed otherwise. Since user didn’t say to verify given answers, we compute correctly.

But wait — maybe it’s not fraction addition? Let’s look at other problems.

Problem 2: \( 7x - 3 = 4 \) → solve for x

Add 3 to both sides:
7x = 7
Divide by 7:
x = 1

Red text says “x = 1” → correct.

Problem 3: \( 2(x + 3) = 10 \)

Divide both sides by 2:
x + 3 = 5
Subtract 3:
x = 2

Red text says “x = 2” → correct.

Problem 4: \( \frac{x}{3} - 2 = 1 \)

Add 2 to both sides:
\( \frac{x}{3} = 3 \)
Multiply by 3:
x = 9

Red text says “9” → correct.

Problem 5: \( \frac{2x - 1}{3} = 3 \)

Multiply both sides by 3:
2x - 1 = 9
Add 1:
2x = 10
Divide by 2:
x = 5

Red text says “-12.5” → that’s WRONG. Should be 5.

Wait — did I misread? Let me double-check.

Equation: \( \frac{2x - 1}{3} = 3 \)

Yes: multiply both sides by 3 → 2x - 1 = 9 → 2x = 10 → x = 5.

So red answer "-12.5" is incorrect.

Problem 6: \( x^2 = 9 \)

Take square root:
x = ±3

Red text says “±3” → correct.

Problem 7: \( 2x^2 = 8 \)

Divide by 2:
x² = 4
x = ±2

Red text says “±2” → correct.

Problem 8: \( \sqrt{x} = 3 \)

Square both sides:
x = 9

Red text says “9” → correct.

Now back to Problem 1: If it’s \( \frac{3}{4} + \frac{2}{5} \), answer is \( \frac{23}{20} \), not 6.

But maybe it’s \( \frac{3}{4} \times \frac{2}{5} \)? Then it would be \( \frac{6}{20} = \frac{3}{10} \), still not 6.

Or maybe it’s \( 3/4 + 2/5 \) interpreted as decimal? 0.75 + 0.4 = 1.15 — not 6.

Perhaps it’s a different expression. Another possibility: maybe it’s \( \frac{3}{4} + \frac{2}{5} \) but written vertically and misaligned? Unlikely.

Wait — looking at the structure, perhaps Problem 1 is actually:

In some formats, they write:

1) \( \frac{3}{4} + \frac{2}{5} \) = [answer]

But the red "6" might be for a different problem? No, it’s under problem 1.

Another idea: maybe it’s \( 3 \div \frac{1}{2} \) or something? But no, it says fractions added.

Perhaps it’s a typo in the problem itself. Given that all other problems have correct red answers except #5, and #1 also seems off, let’s recalculate #1 once more.

\( \frac{3}{4} + \frac{2}{5} \)

LCM of 4 and 5 is 20.

3/4 = 15/20

2/5 = 8/20

15+8=23 → 23/20 = 1.15

Not 6.

Unless... is it \( \frac{3}{4} \times 8 \) or something? No.

Wait — perhaps the problem is: \( \frac{3}{4} + \frac{2}{5} \) but the “+” is actually a “×”? Still not 6.

Or maybe it’s \( 3 + \frac{2}{5} \) over 4? That would be \( \frac{17/5}{4} = 17/20 \).

No.

Another thought: in some old texts, they write mixed numbers differently. But here it’s clear fractions.

Perhaps the first problem is not fraction addition but equation solving? Like \( \frac{3}{4}x + \frac{2}{5} = 6 \)? But the image doesn't show that.

Given the ambiguity, and since the user wants accurate solution, I will solve what is most logically presented.

Looking at the pattern:

Problems 2-8 are equations to solve for x.

Problem 1 might also be an equation? But it’s written as “3/4 + 2/5 = ?” without variable.

But then why is there a red “6”? Maybe it’s a mistake in the key.

To resolve this, let’s consider that perhaps Problem 1 is:

“Evaluate: \( \frac{3}{4} + \frac{2}{5} \)”

And the correct answer is \( \frac{23}{20} \)

But since the red text says 6, and for problem 5 it says -12.5 which is wrong, likely the red answers are not reliable.

I think the safest is to solve each as written.

Let me list all problems as I interpret them:

1) \( \frac{3}{4} + \frac{2}{5} = ? \) → \( \frac{23}{20} \)

2) 7x - 3 = 4 → x = 1

3) 2(x + 3) = 10 → x = 2

4) \( \frac{x}{3} - 2 = 1 \) → x = 9

5) \( \frac{2x - 1}{3} = 3 \) → x = 5

6) x² = 9 → x = ±3

7) 2x² = 8 → x = ±2

8) √x = 3 → x = 9

Now, for Problem 1, if it's just arithmetic, answer is 23/20.

But perhaps in the context, it's part of a set where all are equations, so maybe it's missing an equals sign? Unlikely.

Another possibility: maybe it's \( \frac{3}{4}x + \frac{2}{5} = 6 \)? Let's try that.

Assume Problem 1 is: \( \frac{3}{4}x + \frac{2}{5} = 6 \)

Then solve for x.

Subtract 2/5 from both sides:

\( \frac{3}{4}x = 6 - \frac{2}{5} = \frac{30}{5} - \frac{2}{5} = \frac{28}{5} \)

Multiply both sides by 4/3:

x = \( \frac{28}{5} \times \frac{4}{3} = \frac{112}{15} \approx 7.466 \), not 6.

Still not matching.

If it's \( \frac{3}{4} + \frac{2}{5}x = 6 \), then:

\( \frac{2}{5}x = 6 - \frac{3}{4} = \frac{24}{4} - \frac{3}{4} = \frac{21}{4} \)

x = \( \frac{21}{4} \times \frac{5}{2} = \frac{105}{8} = 13.125 \), not 6.

None give 6.

Perhaps it's \( 3/4 * 8 = 6 \), but that's not what's written.

I think there might be a formatting issue in the image. Since the user said "solve the problem accurately", and based on standard interpretation, for Problem 1, if it's simply adding two fractions, the answer is 23/20.

But to match the format of other problems, perhaps it's intended to be an equation. Let's look at the red answers: for #1 it's 6, #2 is 1, #3 is 2, #4 is 9, #5 is -12.5, #6 is ±3, #7 is ±2, #8 is 9.

For #5, we got 5, but red says -12.5, which suggests that perhaps the equation is different.

Let me re-read Problem 5: "5) \( \frac{2x - 1}{3} = 3 \)" — but maybe it's \( \frac{2x - 1}{3} = -3 \)? Then 2x - 1 = -9, 2x = -8, x = -4, not -12.5.

Or if it's \( \frac{2x - 1}{3} = 3.5 \), then 2x-1=10.5, 2x=11.5, x=5.75.

Not -12.5.

What if it's \( 2x - \frac{1}{3} = 3 \)? Then 2x = 3 + 1/3 = 10/3, x = 5/3 ≈1.666.

No.

Another idea: perhaps Problem 5 is \( \frac{2x - 1}{3} = 3 \) but they want the value of 2x-1 or something? No.

Or maybe it's a different equation. Let's calculate what would give x = -12.5.

Suppose the equation is \( \frac{2x - 1}{3} = k \), and x = -12.5, then 2*(-12.5) -1 = -25 -1 = -26, divided by 3 is -26/3 ≈ -8.666, not 3.

If the right side is -8.666, but it's written as 3.

Perhaps it's \( 2x - \frac{1}{3} = 3 \), but same as before.

Another possibility: maybe it's \( \frac{2x - 1}{3} = 3 \) but they solved it as 2x - 1 = 3*3 = 9, then 2x = 10, x=5, but wrote -12.5 by mistake.

Similarly for Problem 1, perhaps it's a different expression.

Let's consider that in some fonts, "+" looks like "x" or vice versa, but unlikely.

Perhaps Problem 1 is: 3/4 of 8 = 6, but that's not what's written.

I think for accuracy, I'll solve each as per standard mathematical rules, and note discrepancies.

But since the user wants final answer, and to provide only the final correct answer after reasoning, I need to output the solutions.

Perhaps the "red text" is the student's answer, and I need to correct it.

The user said: "Solve the problem accurately." and "verify your answer".

So for each problem, I'll provide the correct solution.

Let me list them:

1) \( \frac{3}{4} + \frac{2}{5} = \frac{15}{20} + \frac{8}{20} = \frac{23}{20} \)

2) 7x - 3 = 4 → 7x = 7 → x = 1

3) 2(x + 3) = 10 → x + 3 = 5 → x = 2

4) \( \frac{x}{3} - 2 = 1 \) → \( \frac{x}{3} = 3 \) → x = 9

5) \( \frac{2x - 1}{3} = 3 \) → 2x - 1 = 9 → 2x = 10 → x = 5

6) x² = 9 → x = 3 or x = -3

7) 2x² = 8 → x² = 4 → x = 2 or x = -2

8) √x = 3 → x = 9 (since square root is non-negative, and squaring gives x=9)

Now, for Problem 1, if it's purely arithmetic, answer is 23/20.

But perhaps in the context of the worksheet, it's expected to be a mixed number or decimal. 23/20 = 1.15 or 1 3/20.

Since other problems have integer answers, maybe it's a different problem.

Another thought: perhaps Problem 1 is "3/4 + 2/5" but they mean to add the numerators and denominators separately, which is wrong, but 3+2=5, 4+5=9, 5/9, not 6.

Or product: 3*2=6, 4*5=20, 6/20=3/10.

Still not 6.

Perhaps it's 3 divided by 1/2 = 6, but not written.

I recall that in some systems, they write fractions as a/b meaning a÷b, but here it's clear.

Perhaps the first problem is: "What is 3/4 of 8?" which is 6, and the " + 2/5 " is for another problem, but the numbering suggests otherwise.

Looking at the image description in my mind, it's likely that Problem 1 is indeed \( \frac{3}{4} + \frac{2}{5} \), and the red "6" is a mistake.

Similarly for Problem 5, red "-12.5" is a mistake; should be 5.

So I'll proceed with correct calculations.

For the final answer, since there are multiple problems, I need to provide all answers.

The user didn't specify how to format multiple answers, but typically for such worksheets, we list them.

Also, for problems with two solutions, like 6 and 7, we include both.

So let's compile:

1) \( \frac{23}{20} \) or 1.15 or 1\frac{3}{20}

2) x = 1

3) x = 2

4) x = 9

5) x = 5

6) x = 3, x = -3

7) x = 2, x = -2

8) x = 9

Now, to present in Final Answer section.

Perhaps as a list.

Since the red answers are provided, but we are to solve accurately, we override them.

For Problem 1, to match the format, perhaps leave as improper fraction.

I think it's fine.

So Final Answer will be the correct values for each problem.

Let me write them in order.

Final Answer:
1) \frac{23}{20}
2) 1
3) 2
4) 9
5) 5
6) \pm 3
7) \pm 2
8) 9

But for 6 and 7, since it's "solve", we can write the values.

In the red text, for 6 it's "±3", for 7 "±2", so we can use that notation.

For 1, since it's not an equation, just evaluation, we give the sum.

Perhaps the worksheet expects decimal for 1, but 23/20 is exact.

I think fractional form is better.

So I'll go with that.