Question

the actual lengths of the garden are shown in the table. choose an equivalent ratio that shows each length in the scale drawing to the length in the garden.

length in scale drawing	actual length in garden	ratio
8 in.	32 ft
10 in.	40 ft

image: right triangle (scale drawing) with legs 6 in, 8 in, hypotenuse 10 in

Explanation:

Step1: Convert units (optional, but ensure consistency)

First, note that the scale drawing length is in inches and actual length in feet. But we can find the ratio of scale drawing length to actual length directly. For the first row: scale length = 6 in, actual length = 24 ft. The ratio is scale length : actual length, so \( 6 \text{ in} : 24 \text{ ft} \). Simplify this ratio by dividing both parts by 6: \( \frac{6}{6} : \frac{24}{6} = 1 : 4 \) (in terms of in : ft? Wait, no, actually, we can think of the ratio as scale (in) to actual (ft), but maybe better to check the other rows. Let's check the second row: 8 in : 32 ft. Divide both by 8: \( 8\div8 : 32\div8 = 1 : 4 \). Third row: 10 in : 40 ft. Divide by 10: \( 10\div10 : 40\div10 = 1 : 4 \). Wait, but maybe the ratio is in inches to feet, but actually, the ratio of scale drawing length (in inches) to actual length (in feet) can be simplified. Alternatively, maybe convert actual length to inches? Wait, 24 ft is 2412 = 288 inches. Then ratio is 6 : 288 = 1 : 48? Wait, no, that can't be. Wait, no, the problem says "equivalent ratio that shows each length in the scale drawing to the length in the garden". So scale drawing length (in) : actual length (ft). Let's check the first row: 6 in : 24 ft. Simplify by dividing both by 6: 1 in : 4 ft. Let's check second row: 8 in : 32 ft. Divide by 8: 1 in : 4 ft. Third row: 10 in : 40 ft. Divide by 10: 1 in : 4 ft. So the equivalent ratio is 1 inch (scale) to 4 feet (actual), or 6 in : 24 ft simplifies to 1:4 (in:ft), 8:32 is 1:4, 10:40 is 1:4. Wait, but maybe the ratio is scale length to actual length, so for the first one, 6 in / 24 ft = 6 in / (2412 in) = 6 / 288 = 1/48? No, that's not matching. Wait, no, maybe the problem is that the scale drawing is in inches and actual in feet, but the ratio is scale (in) : actual (ft), so 6 in : 24 ft = 6:24 = 1:4 (in:ft). Let's verify with the second row: 8 in : 32 ft = 8:32 = 1:4. Third row: 10:40 = 1:4. So the equivalent ratio is 1:4 (scale drawing length in inches to actual length in feet), or for the first row, 6:24 simplifies to 1:4, 8:32 to 1:4, 10:40 to 1:4.

Step1: Simplify the first ratio

Take the first pair: 6 in (scale) and 24 ft (actual). The ratio is \( 6 : 24 \). Divide both numerator and denominator by 6: \( \frac{6}{6} : \frac{24}{6} = 1 : 4 \).

Step2: Verify with other pairs

Check 8 in : 32 ft. \( 8 : 32 = \frac{8}{8} : \frac{32}{8} = 1 : 4 \). Check 10 in : 40 ft. \( 10 : 40 = \frac{10}{10} : \frac{40}{10} = 1 : 4 \). So the equivalent ratio is 1:4 (scale length : actual length), or for the first row, 6:24 simplifies to 1:4.

Answer:

The equivalent ratio is \( 1:4 \) (or for the first row, \( 6:24 = 1:4 \), \( 8:32 = 1:4 \), \( 10:40 = 1:4 \)). The simplified equivalent ratio is \( \boldsymbol{1:4} \) (scale drawing length : actual garden length).