In 2,460, the digit 4 is in the hundreds place (value = 400). Moving it one place to the right puts it in the tens place (value = 40). The new number becomes 2,046, where the 4 represents 40 instead of 400. This confirms the relationship: 400 is 10 times 40. Choice A (2,064) puts 4 in the ones place. Choice B (2,406) rearranges digits incorrectly. Choice D (2,604) moves other digits around incorrectly.

The basic place value relationship states that a digit represents 10 times what it represents in the place to its right. Emma's original equation compared thousands to tens (skipping hundreds), giving a ratio of 100. Choice A correctly compares thousands (7,000) to hundreds (700), showing adjacent places with a ratio of 10. Choice C also shows the 10-to-1 ratio but doesn't fix Emma's specific equation. Choice B gives a ratio of 1,000. Choice D compares tens to ones, which is correct in principle but doesn't address fixing Emma's thousands-place equation.

The digit 8 is in the thousands place, so its value is 8,000. To get 80,000, Carlos multiplied by 10 (since 8,000 × 10 = 80,000). The digit 3 is in the hundreds place, so its value is 300. Dividing by the same number: 300 ÷ 10 = 30. Choice B gives just the digit 3, not its place value divided by 10. Choice C gives the original place value of 3 without dividing. Choice D would result from incorrectly thinking 3 ÷ 10 = 0.3 instead of using the place value 300.

When each digit shifts one place to the right, 45,630 becomes 4,563. This demonstrates that moving a digit one place to the right makes it represent 1/10 of its original value, which is the inverse of the place value relationship. For example, the 4 originally represented 40,000 (ten-thousands place) and now represents 4,000 (thousands place), and 4,000 = 40,000 ÷ 10. Choice A has the relationship backwards. Choice C shows an impossible number format. Choice D focuses on the numerical difference rather than the place value relationship.

When you see numbers with the same digits in different positions, think about how place value affects their relationship. This is a perfect opportunity to explore how our number system works.

Let's examine what happens when we compare $$23,450$$ and $$2,345$$. Notice that $$23,450$$ contains exactly the same digits as $$2,345$$, but each digit has been shifted one place to the left, with a zero added at the end. When we calculate the ratio: $$23,450 ÷ 2,345 = 10$$.

This ratio of 10 reveals a fundamental truth about place value: each position in our number system represents 10 times the value of the position to its right. The 2 in $$23,450$$ is in the ten-thousands place, while the 2 in $$2,345$$ is in the thousands place. Since ten-thousands are worth 10 times more than thousands, the same digit represents 10 times more value.

Choice A incorrectly calculates the difference ($$21,105$$) rather than the ratio. A ratio compares numbers through division, not subtraction. Choice B gives the wrong quotient of 100, likely from misunderstanding how the extra digit affects the calculation. Choice C correctly calculates the ratio as 10 but gives the wrong reasoning about decimal points, which don't apply to whole number place value.

Remember this key insight: when digits shift one place to the left in our base-10 system, their value increases by a factor of 10. This makes ratios a powerful tool for understanding place value relationships.

When you see division patterns like this, focus on how the numbers change while the quotient stays the same. Let's examine what's happening in each step of the given pattern.

In $$50 \div 5 = 10$$, $$500 \div 50 = 10$$, and $$5,000 \div 500 = 10$$, notice that both the dividend (first number) and divisor (second number) are multiplied by 10 each time. When you multiply both numbers in a division problem by the same amount, the answer stays the same. This is why all three equal 10.

Following this pattern, the next step should have $$50,000 \div 5,000 = 10$$. Both numbers are again 10 times larger than the previous step, maintaining the same relationship.

Answer D correctly identifies this pattern. Answer A jumps too far ahead by making both numbers 10 times larger than what the next logical step should be. Answer B breaks the pattern entirely by changing the divisor incorrectly, which gives a quotient of 100 instead of 10. Answer C disrupts the pattern by keeping the divisor as 5 instead of following the sequence where the divisor should be 5,000.

The key insight is recognizing that when both the dividend and divisor increase by the same factor, the quotient remains constant. This demonstrates an important property of division and place value relationships.

Remember: In number patterns involving division, look for how both numbers change together. If they change by the same factor, the answer will stay the same.

This directly applies the place value relationship: a digit in one place represents 10 times what it represents in the place to its right. Here, 9 in the hundreds place (900) represents 10 times what 9 represents in the tens place (90). Choice B incorrectly suggests 900 is 100 times larger than 90. Choice C shows the multiplication pattern but doesn't explain the place value relationship. Choice D incorrectly describes place values as being '10 more' rather than '10 times' what the place to the right represents.

This question tests 4th grade understanding that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (CCSS.4.NBT.1). Our place value system is based on groups of 10. Each place value position is 10 times the position to its right—ones become tens (×10), tens become hundreds (×10), hundreds become thousands (×10). This means that the same digit in adjacent places has values that differ by a factor of 10. Chen compares 700 and 70, where the 7 is in the hundreds place (700) and tens place (70), requiring students to recognize that 700 is 10 times 70. Choice B is correct because calculating 700 ÷ 70 = 10 shows that 700 is 10 times 70. This demonstrates understanding that adjacent place values have a 10-to-1 relationship. Choice A represents using 100 instead of 10 (confused with non-adjacent places), which happens when students don't understand multiplicative relationships between places. To help students: Use place value charts or base-ten blocks to show that 1 hundred = 10 tens, 1 thousand = 10 hundreds. Emphasize the pattern: moving one place to the left multiplies by 10, moving one place to the right divides by 10. Practice with division: 700 ÷ 70 = 10, 5,000 ÷ 500 = 10, 30 ÷ 3 = 10 (always 10 for adjacent places).

This question tests 4th grade understanding that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (CCSS.4.NBT.1). Our place value system is based on groups of 10. Each place value position is 10 times the position to its right—ones become tens (×10), tens become hundreds (×10), hundreds become thousands (×10). This means that the same digit in adjacent places has values that differ by a factor of 10. For example, 7 in the hundreds place (700) is 10 times the 7 in the tens place (70). In the number 2,222, the digit 2 appears in the tens place (value 20) and the ones place (value 2), requiring students to recognize that 20 is 10 times 2. Choice A is correct because calculating that 20 is 10 times 2 demonstrates understanding that adjacent place values have a 10-to-1 relationship. Choice B represents using 100 instead of 10 (confused with non-adjacent places), which happens when students don't understand multiplicative relationships between places. To help students: Use place value charts or base-ten blocks to show that 1 hundred = 10 tens, 1 thousand = 10 hundreds. Emphasize the pattern: moving one place to the left multiplies by 10, moving one place to the right divides by 10. Practice with division: 700 ÷ 70 = 10, 5,000 ÷ 500 = 10, 30 ÷ 3 = 10 (always 10 for adjacent places). Use numbers with repeating digits (4,440, 7,777) to make the relationship clear. Point out that the DIGIT stays the same, but the VALUE changes by 10 times. Watch for: students who subtract instead of divide, students who use 100 for the relationship (that's for places two positions apart), and students who give the digit value instead of the multiplicative relationship.

This question tests 4th grade understanding that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right (CCSS.4.NBT.1). Our place value system is based on groups of 10. Each place value position is 10 times the position to its right—ones become tens (×10), tens become hundreds (×10), hundreds become thousands (×10). This means that the same digit in adjacent places has values that differ by a factor of 10. For example, 7 in the hundreds place (700) is 10 times the 7 in the tens place (70). In the number 6,600, the digit 6 appears in the thousands place (value 6,000) and the hundreds place (value 600), requiring students to recognize that 6,000 is 10 times 600. Choice B is correct because dividing the larger value by the smaller value: 6,000 ÷ 600 = 10, demonstrating understanding that adjacent place values have a 10-to-1 relationship. Choice D represents using 100 instead of 10 (confused with non-adjacent places), which happens when students confuse operations. To help students: Use place value charts or base-ten blocks to show that 1 hundred = 10 tens, 1 thousand = 10 hundreds. Emphasize the pattern: moving one place to the left multiplies by 10, moving one place to the right divides by 10. Practice with division: 700 ÷ 70 = 10, 5,000 ÷ 500 = 10, 30 ÷ 3 = 10 (always 10 for adjacent places). Use numbers with repeating digits (4,440, 7,777) to make the relationship clear. Point out that the DIGIT stays the same, but the VALUE changes by 10 times. Watch for: students who subtract instead of divide, students who use 100 for the relationship (that's for places two positions apart), and students who give the digit value instead of the multiplicative relationship.