Let's break down the statements to clarify the concepts of tens and tenths and why \(3 \times \frac{1}{10}\) is less than \(3 \times 10\).
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Understanding the terms:
- "Ten" refers to the whole number 10.
- "Tenth" refers to one part out of ten parts, which is represented as \(\frac{1}{10}\) or 0.1.
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Calculating values:
- \(3 \times 10 = 30\).
- \(3 \times \frac{1}{10} = 3 \times 0.1 = 0.3\).
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Comparison:
- \(30\) (which is \(3 \times 10\)) is far greater than \(0.3\) (which is \(3 \times \frac{1}{10}\)).
Now, let's analyze the statements:
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"The value of 1 ten is less than the value of 3 tens."
- This statement is true, as \(10\) is less than \(30\).
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"The value of 1 tenth is less than the value of 3 tenths."
- This is also true, since \(0.1\) (1 tenth) is less than \(0.3\) (3 tenths).
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"The value of 1 tenth is less than the value of 1 ten."
- This statement is true, as \(0.1\) (1 tenth) is indeed less than \(10\) (1 ten).
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"The value of 1 ten is less than the value of 1 tenth."
- This statement is false; \(10\) (1 ten) is not less than \(0.1\) (1 tenth). In fact, it's much greater.
Summary:
The comparison \(3 \times \frac{1}{10}\) is less than \(3 \times 10\) because the individual values of a tenth (0.1) are drastically lower than that of a ten (10). Therefore, when you multiply each by 3, you maintain that relationship.