Since area = length × width, we have $$\frac{15}{8} = length × \frac{3}{4}$$. To find length, divide: $$length = \frac{15}{8} ÷ \frac{3}{4} = \frac{15}{8} × \frac{4}{3} = \frac{60}{24} = \frac{5}{2}$$ feet. Choice A incorrectly subtracts. Choice C incorrectly multiplies the area by width. Choice D incorrectly adds the area and width.

When you see a rectangle tiled with unit squares, you need to connect two different ways of finding area: counting the squares directly and multiplying the rectangle's side lengths.

Let's work through this step by step. You have 15 squares along the length and 8 squares along the width, so counting squares gives you $$15 × 8 = 120$$ total squares. Since each square has area $$\frac{1}{10} × \frac{1}{10} = \frac{1}{100}$$ square inches, the total area is $$120 × \frac{1}{100}$$ square inches.

Now for the side lengths: the length is $$15 × \frac{1}{10} = \frac{15}{10}$$ inches, and the width is $$8 × \frac{1}{10} = \frac{8}{10}$$ inches. Multiplying these gives area = $$\frac{15}{10} × \frac{8}{10}$$ square inches.

Choice C correctly shows both methods are equal: $$120 × \frac{1}{100} = \frac{15}{10} × \frac{8}{10}$$.

Choice A uses $$\frac{1}{10}$$ instead of $$\frac{1}{100}$$ for the unit square area, forgetting that area requires squaring the side length. Choice B tries to add the side lengths instead of multiplying them, and also uses the wrong unit square area. Choice D makes two errors: it uses $$\frac{1}{100}$$ in the counting method (should be $$\frac{1}{100}$$) and incorrectly represents the side lengths as $$\frac{15}{100}$$ and $$\frac{8}{100}$$.

Remember: when unit squares have side length $$s$$, their area is $$s^2$$, not just $$s$$. Always square the side length when finding the area of each unit square.

The core skill is finding rectangular areas with fractional sides, like 2/3 yard by 3/4 yard, via fraction multiplication. Partitioning into unit fraction squares of 1/3 yard by 1/4 yard creates a 2 by 3 grid of 6 small squares. The multiplication link is 2/3 times 3/4 equaling 1/2 square yard, matching 6 times 1/12 square yard. Square units, such as square yards, denote the area coverage. One misconception is adding fractions for area, like 2/3 + 3/4 = 17/12, which wrongly confuses it with linear addition. Such tiling models bolster the area formula by contrasting correct multiplication with errors. Overall, they generalize the support for formulas, showing how visuals prevent misconceptions in fractional geometry.

Total dimensions: length = $$4 × \frac{1}{6} = \frac{4}{6} = \frac{2}{3}$$ yard, width = $$3 × \frac{1}{8} = \frac{3}{8}$$ yard. Total area = $$\frac{2}{3} × \frac{3}{8} = \frac{6}{24} = \frac{1}{4}$$ square yard. Choice A incorrectly adds $$\frac{4}{6} + \frac{3}{8}$$. Choice B represents $$12 × \frac{1}{48}$$ from multiplying piece count by wrong unit area. Choice C uses incorrect fraction arithmetic.

When you see a problem about cutting paper into strips, you need to figure out what measurement determines how many strips you can make. The key is understanding which dimension matters for the cutting.

Since Kevin cuts the paper into strips that are $$\frac{1}{4}$$ foot wide, he's cutting across the width of the rectangle. To find how many strips he gets, you need to divide the width of the original paper by the width of each strip.

First, find the original width using the area formula. Since Area = length × width, we have $$\frac{5}{6} = \frac{5}{3} \times \text{width}$$. Solving for width: $$\text{width} = \frac{5}{6} ÷ \frac{5}{3} = \frac{5}{6} \times \frac{3}{5} = \frac{1}{2}$$ foot.

Now divide the width by the strip width: $$\frac{1}{2} ÷ \frac{1}{4} = \frac{1}{2} \times \frac{4}{1} = 2$$ strips. Choice A is correct.

Choice B incorrectly divides the total area by the strip width, but area doesn't tell you how many strips you can cut. Choice C divides the length by the strip width, but since he's cutting across the width (not along the length), the length doesn't determine the number of strips. Choice D multiplies instead of dividing, which would give you a portion of a strip rather than counting strips.

Remember: When cutting strips, always identify which dimension is being divided. If strips are a certain width, divide the paper's width by the strip width to find how many strips you get.

First, find the dimensions: length = 6 × $$\frac{1}{4}$$ = $$\frac{6}{4}$$ feet, width = 8 × $$\frac{1}{4}$$ = $$\frac{8}{4}$$ feet. Area = $$\frac{6}{4} × \frac{8}{4} = \frac{48}{16}$$ square feet. Choice A adds the dimensions instead of multiplying. Choice C incorrectly calculates 6 × 4 in the numerator. Choice D uses the wrong denominator.

The core skill is finding the area of a rectangle with fractional side lengths by multiplying the length and width directly. We can tile the rectangle with unit fraction squares that are 1/4 foot by 1/3 foot, each covering 1/12 square foot, and here the 3/4-foot by 2/3-foot rectangle fits exactly 6 such tiles without gaps or overlaps. The total area from tiling, which is 6 times 1/12 or 1/2 square foot, connects directly to multiplying the side lengths 3/4 by 2/3 to get the same 1/2 square foot. The area is expressed in square feet, meaning the space covered as if using 1-foot by 1-foot units, but fractional tiles help visualize partial units. A common misconception is that you need whole-number sides to calculate area, but fractions work just as well by multiplying numerators and denominators. Visual models like this tiling demonstrate how the area formula length times width applies even to fractions. These models generalize to support that area formulas hold for any rational dimensions, building intuition for more complex shapes.

The core skill is computing areas of rectangles with fractional sides, such as 3/4 meter by 2/3 meter, by multiplying the fractions. Tiling with unit fraction squares of 1/4 meter by 1/3 meter fits 3 by 2 for 6 small squares, visualizing the area. This connects to multiplication since 3/4 times 2/3 is 1/2 square meter, equaling the total from 6 squares of 1/12 each. Square units like square meters express the two-dimensional measure. A key misconception addressed here is adding sides for area, as in claiming 3/4 + 2/3 = 17/12, which is incorrect and actually relates to perimeter. Tiling models support the area formula by demonstrating why multiplication, not addition, is used. In general, these visuals generalize how formulas derive from countable units, clarifying errors in area calculation.

The core skill is finding the area of a rectangle with fractional sides, such as a sticker that is 1/2 inch long and 3/4 inch wide, by multiplying those lengths to get 3/8 square inch. Tiling the rectangle with unit fraction squares that are 1/4 inch by 1/4 inch helps visualize how the space is covered without gaps or overlaps. Counting the tiles shows there are 6 such squares, each with area 1/16 square inch, totaling 3/8 square inch, which connects directly to multiplying 1/2 by 3/4. The area is measured in square inches, where each square inch represents a 1 inch by 1 inch unit, but fractions allow for partial units. A common misconception is that areas with fractional sides must be whole numbers, but tiling demonstrates that fractional areas are valid and precise. Models like tiling build intuition for why the area formula works with fractions. These visual aids generalize to support the formula area equals length times width for any real numbers.

The core skill is finding the area of a rectangle with fractional sides, such as a notebook cover that is 1/2 foot long and 2/3 foot wide, by multiplying those lengths to get 1/3 square foot. Tiling the rectangle with unit fraction squares that are 1/6 foot by 1/6 foot helps visualize how the space is covered without gaps or overlaps. Counting the tiles shows there are 12 such squares, each with area 1/36 square foot, totaling 1/3 square foot, which connects directly to multiplying 1/2 by 2/3. The area is measured in square feet, where each square foot represents a 1 foot by 1 foot unit, but fractions allow for partial units. A common misconception is that areas with fractional sides must be whole numbers, but tiling demonstrates that fractional areas are valid and precise. Models like tiling build intuition for why the area formula works with fractions. These visual aids generalize to support the formula area equals length times width for any real numbers.