Math worksheet featuring six problems that test understanding of equality properties without performing calculations.

Problem Analysis

The worksheet asks us to analyze given equalities and determine which other equalities must also be true based on the properties of equality and proportions. We will solve each problem step by step, using logical reasoning and mathematical principles.

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Problem 1

Given:
\[
\frac{264}{284} = \frac{4488}{4828}
\]

We need to determine which of the following must also be true:
1. \( 264 + 4828 = 284 + 4488 \)
2. \( \frac{264^2}{284^2} = \frac{4488^2}{4828^2} \)
3. \( 264 \cdot 4828 = 284 \cdot 4488 \)
4. \( 264 \cdot 4488 = 284 \cdot 4828 \)
5. \( \frac{264}{4828} = \frac{4488}{284} \)

#### Solution:
- Property of Proportions: If \(\frac{a}{b} = \frac{c}{d}\), then:
1. \( ad = bc \) (Cross-multiplication)
2. \( \frac{a^2}{b^2} = \frac{c^2}{d^2} \) (Squaring both sides)
3. \( \frac{a}{d} = \frac{c}{b} \) (Inverting the fractions)

Using these properties:
1. \( 264 + 4828 = 284 + 4488 \) is not necessarily true because addition does not preserve proportionality.
2. \( \frac{264^2}{284^2} = \frac{4488^2}{4828^2} \) is true because squaring both sides of a proportion maintains the equality.
3. \( 264 \cdot 4828 = 284 \cdot 4488 \) is true because it is the cross-multiplication of the original proportion.
4. \( 264 \cdot 4488 = 284 \cdot 4828 \) is false because it reverses the correct cross-multiplication.
5. \( \frac{264}{4828} = \frac{4488}{284} \) is false because inverting the fractions incorrectly swaps the numerators and denominators.

Correct Answer:
\[
\boxed{\frac{264^2}{284^2} = \frac{4488^2}{4828^2}, \quad 264 \cdot 4828 = 284 \cdot 4488}
\]

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Problem 2

Given:
\[
\frac{297}{83} = \frac{2673}{747}
\]

We need to determine which of the following must also be true:
1. \( \frac{297}{747} = \frac{2673}{83} \)
2. \( 297 \cdot 747 = 83 \cdot 2673 \)
3. \( 297 + 747 = 83 + 2673 \)
4. \( 297 \cdot 2673 = 83 \cdot 747 \)
5. \( \frac{297^2}{83^2} = \frac{2673^2}{747^2} \)

#### Solution:
Using the same properties of proportions:
1. \( \frac{297}{747} = \frac{2673}{83} \) is false because inverting the fractions incorrectly swaps the numerators and denominators.
2. \( 297 \cdot 747 = 83 \cdot 2673 \) is true because it is the cross-multiplication of the original proportion.
3. \( 297 + 747 = 83 + 2673 \) is not necessarily true because addition does not preserve proportionality.
4. \( 297 \cdot 2673 = 83 \cdot 747 \) is false because it reverses the correct cross-multiplication.
5. \( \frac{297^2}{83^2} = \frac{2673^2}{747^2} \) is true because squaring both sides of a proportion maintains the equality.

Correct Answer:
\[
\boxed{297 \cdot 747 = 83 \cdot 2673, \quad \frac{297^2}{83^2} = \frac{2673^2}{747^2}}
\]

---

Problem 3

Given:
\[
\frac{146}{23} = \frac{2774}{437}
\]

We need to determine which of the following must also be true:
1. \( \frac{146^2}{23^2} = \frac{2774^2}{437^2} \)
2. \( \frac{146}{437} = \frac{2774}{23} \)
3. \( 146 + 437 = 23 + 2774 \)
4. \( 146 \cdot 437 = 23 \cdot 2774 \)
5. \( 146 \cdot 2774 = 23 \cdot 437 \)

#### Solution:
Using the properties of proportions:
1. \( \frac{146^2}{23^2} = \frac{2774^2}{437^2} \) is true because squaring both sides of a proportion maintains the equality.
2. \( \frac{146}{437} = \frac{2774}{23} \) is false because inverting the fractions incorrectly swaps the numerators and denominators.
3. \( 146 + 437 = 23 + 2774 \) is not necessarily true because addition does not preserve proportionality.
4. \( 146 \cdot 437 = 23 \cdot 2774 \) is true because it is the cross-multiplication of the original proportion.
5. \( 146 \cdot 2774 = 23 \cdot 437 \) is false because it reverses the correct cross-multiplication.

Correct Answer:
\[
\boxed{\frac{146^2}{23^2} = \frac{2774^2}{437^2}, \quad 146 \cdot 437 = 23 \cdot 2774}
\]

---

Problem 4

Given:
\[
\frac{240}{272} = \frac{9120}{10336}
\]

We need to determine which of the following must also be true:
1. \( 240 + 10336 = 272 + 9120 \)
2. \( 240 \cdot 9120 = 272 \cdot 10336 \)
3. \( \frac{240}{10336} = \frac{9120}{272} \)
4. \( \frac{240^2}{272^2} = \frac{9120^2}{10336^2} \)
5. \( 240 \cdot 10336 = 272 \cdot 9120 \)

#### Solution:
Using the properties of proportions:
1. \( 240 + 10336 = 272 + 9120 \) is not necessarily true because addition does not preserve proportionality.
2. \( 240 \cdot 9120 = 272 \cdot 10336 \) is true because it is the cross-multiplication of the original proportion.
3. \( \frac{240}{10336} = \frac{9120}{272} \) is false because inverting the fractions incorrectly swaps the numerators and denominators.
4. \( \frac{240^2}{272^2} = \frac{9120^2}{10336^2} \) is true because squaring both sides of a proportion maintains the equality.
5. \( 240 \cdot 10336 = 272 \cdot 9120 \) is false because it reverses the correct cross-multiplication.

Correct Answer:
\[
\boxed{240 \cdot 9120 = 272 \cdot 10336, \quad \frac{240^2}{272^2} = \frac{9120^2}{10336^2}}
\]

---

Problem 5

Given:
\[
\frac{286}{112} = \frac{18304}{7168}
\]

We need to determine which of the following must also be true:
1. \( 286 + 7168 = 112 + 18304 \)
2. \( 286 \cdot 18304 = 112 \cdot 7168 \)
3. \( \frac{286}{7168} = \frac{18304}{112} \)
4. \( \frac{286^2}{112^2} = \frac{18304^2}{7168^2} \)
5. \( 286 \cdot 7168 = 112 \cdot 18304 \)

#### Solution:
Using the properties of proportions:
1. \( 286 + 7168 = 112 + 18304 \) is not necessarily true because addition does not preserve proportionality.
2. \( 286 \cdot 18304 = 112 \cdot 7168 \) is false because it reverses the correct cross-multiplication.
3. \( \frac{286}{7168} = \frac{18304}{112} \) is false because inverting the fractions incorrectly swaps the numerators and denominators.
4. \( \frac{286^2}{112^2} = \frac{18304^2}{7168^2} \) is true because squaring both sides of a proportion maintains the equality.
5. \( 286 \cdot 7168 = 112 \cdot 18304 \) is true because it is the cross-multiplication of the original proportion.

Correct Answer:
\[
\boxed{\frac{286^2}{112^2} = \frac{18304^2}{7168^2}, \quad 286 \cdot 7168 = 112 \cdot 18304}
\]

---

Problem 6

Given:
\[
122 + 286 = 281 + 127
\]

We need to determine which of the following must also be true:
1. \( 122 - 127 = 281 - 286 \)
2. \( 286 - 281 = 127 - 122 \)
3. \( \frac{122}{127} = \frac{281}{286} \)
4. \( 122 \cdot 127 = 286 \cdot 281 \)
5. \( 122 + 127 = 286 + 281 \)

#### Solution:
The given equation is an equality of sums. We can manipulate it algebraically:
\[
122 + 286 = 281 + 127 \implies 122 - 127 = 281 - 286
\]

Using this:
1. \( 122 - 127 = 281 - 286 \) is true because it is a direct rearrangement of the given equation.
2. \( 286 - 281 = 127 - 122 \) is true because it is the negation of the previous equality.
3. \( \frac{122}{127} = \frac{281}{286} \) is not necessarily true because the given equation does not imply proportionality.
4. \( 122 \cdot 127 = 286 \cdot 281 \) is not necessarily true because multiplication does not follow from the given sum equality.
5. \( 122 + 127 = 286 + 281 \) is false because it contradicts the given equation.

Correct Answer:
\[
\boxed{122 - 127 = 281 - 286, \quad 286 - 281 = 127 - 122}
\]

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Final Answers

1. \(\boxed{\frac{264^2}{284^2} = \frac{4488^2}{4828^2}, \quad 264 \cdot 4828 = 284 \cdot 4488}\)
2. \(\boxed{297 \cdot 747 = 83 \cdot 2673, \quad \frac{297^2}{83^2} = \frac{2673^2}{747^2}}\)
3. \(\boxed{\frac{146^2}{23^2} = \frac{2774^2}{437^2}, \quad 146 \cdot 437 = 23 \cdot 2774}\)
4. \(\boxed{240 \cdot 9120 = 272 \cdot 10336, \quad \frac{240^2}{272^2} = \frac{9120^2}{10336^2}}\)
5. \(\boxed{\frac{286^2}{112^2} = \frac{18304^2}{7168^2}, \quad 286 \cdot 7168 = 112 \cdot 18304}\)
6. \(\boxed{122 - 127 = 281 - 286, \quad 286 - 281 = 127 - 122}\)