The original 7 has value 0.7. We want (1/100) × 0.7 = 0.007. This means the new 7 should be in the thousandths place. Moving two places to the right decreases the value by 1/100, since each place represents 1/10 of the previous place.

When you encounter place value problems involving very large or small numbers, focus on understanding what each digit's position actually represents in terms of value.

Let's think through this step by step. If a digit $$d$$ is in the tens place, its value is $$10d$$. If the same digit $$d$$ is in the thousandths place, its value is $$0.001d$$ (since thousandths means $$\frac{1}{1000}$$). Ms. Chen claims the tens place value is 10,000 times larger than the thousandths place value.

To verify this relationship, we need: $$10d = 10{,}000 \times 0.001d$$. Let's check: $$10{,}000 \times 0.001 = 10$$, so $$10{,}000 \times 0.001d = 10d$$. This confirms the relationship is correct.

Choice A incorrectly uses $$0.01d$$ (hundredths) instead of $$0.001d$$ (thousandths), and uses 100 instead of 10 for the tens place value. Choice B makes the same error with hundredths place and incorrectly uses 1,000 as the multiplier instead of 10,000. Choice D uses $$0.1d$$ (tenths place) rather than thousandths, and the multiplier 100 is far too small.

Only choice C correctly represents both place values: $$10d$$ for tens place and $$0.001d$$ for thousandths place, with the accurate 10,000 multiplier.

Remember: When comparing place values, always convert the position to its decimal equivalent first (tens = ×10, thousandths = ×0.001), then check if the multiplication relationship holds true.

The value of a digit in a number depends on its position or place in the number. Each place value is 10 times greater than the place immediately to its right. Similarly, each place value is 1/10 of the place immediately to its left. For example, in the number 23.45, the digit 4 in the tenths place has a value of 0.4, which is 10 times the value of the digit 5 in the hundredths place at 0.05. A common misconception is that digits have fixed values regardless of position, but position multiplies the digit by the place's value, like tenths being 0.1. Understanding place value relationships allows us to compare digits across positions accurately. This knowledge helps in reading, writing, and operating on decimal numbers effectively.

When you see a question about place value, remember that each digit's value depends on its position in the number. The digit itself stays the same, but its actual value changes based on where it sits.

Let's work through this step by step. You have the digit $$8$$ in three places: ones, hundredths, and ten-thousandths. To find each digit's value, you multiply the digit by its place value:

• The $$8$$ in the ones place: $$8 \times 1 = 8$$
• The $$8$$ in the hundredths place: $$8 \times 0.01 = 0.08$$
• The $$8$$ in the ten-thousandths place: $$8 \times 0.0001 = 0.0008$$

Adding these values together: $$8 + 0.08 + 0.0008 = 8.0808$$

Answer choice D correctly shows this calculation and gives the right sum.

Now let's see why the other answers miss the mark. Choice A gives $$8.08008$$, which incorrectly places the ten-thousandths value in the hundred-thousandths place. Choice B suggests $$8.888$$, falling into the trap of thinking each $$8$$ has equal value regardless of position—this ignores how place value works. Choice C calculates $$24$$ by simply multiplying $$3 \times 8$$, completely overlooking that place value determines actual worth, not just counting digits.

Study tip: When working with place value, always convert each digit to its actual value first, then add. Don't get fooled by how many times a digit appears—focus on where each digit sits in the number.

The digit 3 in the tenths place represents 0.3. If moved to the hundredths place, it would represent 0.03. Since 0.03 = (1/10) × 0.3, the new value is 1/10 of the original value. This demonstrates that each place to the right represents 1/10 of the place to its left.

When comparing decimal place values, you need to understand how each position relates to the others. Each place value is exactly 10 times larger than the position to its right.

Let's examine what each 9 actually represents. In $$52.39$$, the 9 is in the hundredths place, so it represents $$9 \times 0.01 = 0.09$$. In $$523.9$$, the 9 is in the tenths place, so it represents $$9 \times 0.1 = 0.9$$.

To find how many times larger the second 9 is, divide: $$0.9 \div 0.09 = 10$$. The 9 in $$523.9$$ is only 10 times larger than the 9 in $$52.39$$, making Student B incorrect.

Looking at the wrong answers: Choice A incorrectly states that tenths are 100 times larger than hundredths, when they're actually only 10 times larger. Choice B makes an even bigger error, claiming the difference is 1000 times. Choice C repeats the same mistake as A, incorrectly stating the relationship is exactly 100 times. All of these reflect a common misconception about decimal place value relationships.

Remember this pattern: each decimal place is exactly 10 times the value of the place to its right. Tenths are 10 times hundredths, hundredths are 10 times thousandths, and so on. When you move one place to the left, you multiply by 10, not 100. This consistent "times 10" relationship is key to mastering decimal comparisons.

A = 5 (ones place), B = 0.5 (tenths place), C = 50 (tens place). Since C is in the tens place and A is in the ones place, C = 10A. Since A is in the ones place and B is in the tenths place, A = 10B. Therefore C = 100B.

In 6.847, the 4 represents 4 hundredths (0.04). To make it 10 times greater, it needs to represent 0.4 (4 tenths). Moving one place to the left (from hundredths to tenths) increases the value by a factor of 10.

When you encounter a problem about moving digits in a decimal number, you're working with place value concepts. The key is to understand that moving digits to the right decreases their place values, while moving them left increases their place values.

Starting with $$1{,}234.567$$, let's trace what happens when each digit moves exactly two places to the right. The digit $$1$$ (in the thousands place) moves to the tens place. The $$2$$ (in the hundreds place) moves to the ones place. The $$3$$ (in the tens place) moves to the tenths place. The $$4$$ (in the ones place) moves to the hundredths place. The $$5$$ (in the tenths place) moves to the thousandths place. The $$6$$ and $$7$$ would move to the ten-thousandths and hundred-thousandths places respectively. This gives us $$12.34567$$.

Answer A is incorrect because it suggests the decimal point moves, but the problem specifically states that digits move, not the decimal point. The decimal point stays in its original position as a reference.

Answer B is wrong because it places all digits after the decimal point, which would require moving them much more than two places to the right. This represents a misunderstanding of how place value shifting works.

Answer D gives the same numerical result as C, but the reasoning is flawed. The digits don't maintain their relative positions to each other—they all shift uniformly by the same amount.

Remember: when digits move right, their values decrease; when they move left, their values increase. Always count place values carefully to avoid errors.

The value of a digit in a decimal number depends on its position or place relative to the decimal point. Each place to the left of another is 10 times greater in value than the place to its right. Conversely, each place to the right is 1/10 the value of the place to its left. For example, in 62,718.043, the 4 in the hundredths place is 4 × 0.01 = 0.04, and the 3 in the thousandths place is 3 × 0.001 = 0.003, so 0.04 is more than 10 times 0.003 (actually about 13.33 times) because 4 > 3. A common misconception is that place value relationships always yield exactly 10 times between adjacent places, regardless of digit values. Understanding place value enables us to compare digits' contributions across positions, even when they differ. This helps us comprehend the structure of decimals, improving skills in comparison and arithmetic.