Question

Tell the nearest tenths of a centimeter that each given measurement is between. For example, 3.66 is between 3.6 and 3.7 \(1.01 \mathrm{cm}\)

Short answer

1.01 cm is between 1.0 cm and 1.1 cm.

Step-by-step solution

Identify the tenths place

Look at the given measurement: 1.01 cm. Identify the digit in the tenths place, which is the first digit after the decimal point. Here, the tenths place is '0'.

Determine the lower bound

The lower bound is formed by taking the tenths place digit and decimal value and then rounding down to the nearest tenth. So, from 1.01 cm, the lower bound is 1.0 cm.

Determine the upper bound

The upper bound is one tenth higher than the lower bound. Add 0.1 to the lower bound. Thus, 1.0 cm + 0.1 cm = 1.1 cm. So, the upper bound is 1.1 cm.

Construct the interval

The measurement 1.01 cm is between the lower bound, 1.0 cm, and the upper bound, 1.1 cm.

Key concepts

Understanding Place Value

Place value is a fundamental concept in mathematics that helps us understand the value of each digit in a number based on its position. In the decimal number system, each place represents a power of 10. For instance, in the number 1.01, each digit has a specific place value:

• The 1 before the decimal point is in the 'ones' place (10^0), so it represents 1.
• The 0 after the decimal point is in the 'tenths' place (10^-1), so it represents 0.0.
• The last 1 is in the 'hundredths' place (10^-2), so it represents 0.01.

Understanding these values helps us perform rounding and comparison of numbers with greater precision. The place value is pivotal in identifying which digit to consider when rounding off measurements or performing operations like addition and subtraction with decimals.

The Role of Decimal Points

A decimal point is a dot used to separate the whole number part from the fractional part of a number. In measurements and calculations, the decimal point allows more precise values. In the example 1.01 cm, the decimal point separates the whole number 1 from the fractional part 0.01. This placement enables us to specify measurements between whole numbers with high accuracy. When rounding numbers, the decimal point helps in identifying which digit to analyze for the rounding process. For instance, to round 1.01 cm to the nearest tenth, you look at the digit in the hundredths place. If this digit is 5 or higher, you round up the digit in the tenths place; otherwise, you round down. The decimal point thus ensures clarity and accuracy in measurements and calculations.

Understanding Intervals

An interval represents a range of values between two numbers. In the context of rounding measurements, intervals help form the boundary within which a given number falls. To define an interval, you need a lower bound and an upper bound:

• The lower bound is the smallest number in the interval, determined by rounding down to the nearest specified unit (like tenths).
• The upper bound is the largest number in the interval, found by adding a specific value (like 0.1) to the lower bound.

For example, for the measurement 1.01 cm, identifying that it is between 1.0 cm and 1.1 cm defines the interval within which 1.01 cm falls. The concept of intervals is widely used in various fields, from mathematics to science, to understand and describe ranges of data or measurements accurately.