Question

Shade the given percent of a 100 -grid. Then express the shaded portion as a fraction and a decimal. $$72 \%$$

Short answer

Shade 72 squares in the 100-grid. The fraction is \( \frac{18}{25} \). The decimal is 0.72.

Step-by-step solution

Shade the Grid

Begin by taking a 100-grid. This grid is a 10x10 square, making a total of 100 small squares. Since the task is to shade 72%, shade 72 out of the 100 small squares. This could be done by shading 7 full rows (which is 70 squares) and 2 additional squares in the 8th row.

Calculate the Fraction

Next, we express the shaded portion as a fraction. Since 72 out of the 100 squares are shaded, this can be written as the fraction \ \( \frac{72}{100} \ \). This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor (4): \ \( \frac{72 \div 4}{100 \div 4} = \frac{18}{25} \ \).

Calculate the Decimal

Finally, convert the shaded portion into a decimal. Since percentages are out of 100, 72% directly translates to 0.72 as a decimal (72 ÷ 100 = 0.72).

Key concepts

Fractions

Fractions represent parts of a whole. In the context of a 100-grid, it is easy to visualize how parts make up the whole grid. For example, when we have 72 out of 100 squares shaded, we can write this as the fraction \( \frac{72}{100} \). This fraction can often be simplified. To simplify \( \frac{72}{100} \), find the greatest common divisor (GCD) of 72 and 100, which is 4. Divide the numerator and the denominator by this GCD: \( \frac{72 \div 4}{100 \div 4} = \frac{18}{25} \). Simplified fractions represent the same value but are easier to understand and work with in many cases.

Decimals

Decimals are another way to represent parts of a whole. A decimal point separates whole numbers from their fractional parts. For example, in the percentage 72%, it can be directly converted into a decimal form by dividing the percentage by 100. Therefore, 72% becomes 0.72 (72 ÷ 100 = 0.72). Decimals are useful in many areas of science, engineering, and everyday life because they make arithmetic calculations simpler. Unlike fractions, decimals are especially easy to read and compare. For instance, knowing that 0.72 is greater than 0.5 is often simpler than comparing \( \frac{72}{100} \) and \( \frac{50}{100} \).

Hundreds Grid

A hundreds grid, typically a 10x10 square, is a powerful visual aid for understanding percentages, fractions, and decimals. It breaks a whole into 100 equal parts, making it easier to see what portion of the whole is represented by shaded squares. For example, to represent 72%, you shade 72 out of 100 squares. This shaded portion demonstrates that \( \frac{72}{100} \) squares are filled, equivalent to 0.72 as a decimal. Visuals like this help students better grasp these concepts because they can see and count the parts of the whole. Using a hundreds grid is particularly useful for learning because it provides a clear, concrete representation of abstract numerical ideas.