Scale factors and scale drawings

1. The scale says 1 inch on the map = 20 real miles.
2. The towns are 3.5 inches apart, so multiply: 3.5 \times 20 = 70

The towns are 70 miles apart.

Check: 3 inches is 3 \times 20 = 60 miles and half an inch is 10 miles; 60 + 10 = 70 ✓

Tip: Map → real life means multiply by the miles per inch. Real distances are always bigger than map distances, so any answer smaller than 3.5 (like 5.7) or barely bigger (like 23.5) can be eliminated quickly.

1. The scale is 1 inch = 4 feet.
2. The width on the plan is 3.5 inches. Plan → real life, so multiply: 3.5 \times 4 = 14

The living room is 14 feet wide.

Check: Divide back: 14 \div 4 = 3.5 in on the plan ✓ (The length is 5 \times 4 = 20 ft, and a 20 ft by 14 ft room is a sensible living-room size.)

Tip: On a floor plan, use the dimension the question asks for. Read the labels carefully: the 5-in side is the length, and the 3.5-in side is the width.

1. Each map inch stands for 2.5 miles, so multiply the map length by 2.5: 4.4 \times 2.5
2. Break it up: 4 \times 2.5 = 10 and 0.4 \times 2.5 = 1.
3. Add: 10 + 1 = 11 miles.

The actual trail is 11 miles long.

Check: Proportion: \frac{1\text{ in}}{2.5\text{ mi}} = \frac{4.4\text{ in}}{11\text{ mi}}, because 2.5 \times 4.4 = 11 ✓

Tip: Map length → real length: multiply by the miles per inch. Real lengths are almost always much bigger than map lengths, so an answer smaller than 4.4 miles is a red flag.

1. Scale factor = new length ÷ original length: k = \frac{9\text{ in}}{36\text{ in}}
2. Simplify by dividing top and bottom by 9: \frac{9}{36} = \frac{1}{4}

The scale factor is \tfrac{1}{4}.

Check: 36 \times \tfrac{1}{4} = 9 inches ✓

Tip: Always put the new figure on top: k = \dfrac{\text{new}}{\text{original}}. A reduction has a scale factor less than 1; an enlargement has a scale factor greater than 1. If you got 4 for a reduction, you divided the wrong way.

1. Set up a proportion of height to shadow (same order on both sides): \frac{6\text{ ft}}{4\text{ ft}} = \frac{h}{22\text{ ft}}
2. Each object's height is \tfrac{6}{4} = 1.5 times its shadow.
3. h = 1.5 \times 22 = 33\text{ ft}

The tree is 33 feet tall.

Check: \frac{33}{22} = 1.5 and \frac{6}{4} = 1.5 — the ratios match ✓

Tip: In similar figures, matching sides are related by multiplying by the same factor, not by adding the same amount. Keep the same order (height over shadow) in both fractions.

1. Scale: 1 inch = 40 miles.
2. Real → map, so divide by the miles per inch: \frac{150\text{ mi}}{40\ \text{mi per in}} = 3.75\text{ in}

The highway is 3.75 inches long on the map.

Check: 3.75 \times 40 = 150 miles ✓

Tip: Set up a proportion to keep the direction straight: \dfrac{1\text{ in}}{40\text{ mi}} = \dfrac{x\text{ in}}{150\text{ mi}}, so x = \dfrac{150}{40} = 3.75. Keeping inches on top on both sides prevents an upside-down answer like 0.27.

1. Convert the real length to inches: 15\text{ ft} \times 12\ \tfrac{\text{in}}{\text{ft}} = 180\text{ in}
2. The model is \tfrac{1}{24} the size of the real car: 180 \div 24 = 7.5\text{ in}

The model is 7.5 inches long.

Check: 7.5 \times 24 = 180 in, and 180 \div 12 = 15 ft ✓

Tip: 15 \div 24 = 0.625 is correct only in feet (0.625 ft). Because the answer is asked in inches, convert before (or after) using the scale. A model should be small — answers like 360 or 4,320 inches go in the wrong direction.

1. Total map distance: 2.25 + 1.5 = 3.75\text{ in}
2. Scale: 1 inch = 12 miles. Map → real, so multiply: 3.75 \times 12 = 45\text{ mi}

Jamal drives 45 miles.

Check: Leg by leg: 2.25 \times 12 = 27 mi and 1.5 \times 12 = 18 mi; 27 + 18 = 45 ✓

Tip: With several legs, you can add the map lengths first and multiply once, or convert each leg and add — the distributive property guarantees the same total. Read the question for every leg of the trip so you don't stop after one road.

1. Total map distance: 1.75 + 1.25 = 3\text{ in}
2. The scale is \tfrac{1}{2} inch = 3 miles, so 1 inch = 6 miles (double both parts).
3. Actual distance: 3\text{ in} \times 6\ \tfrac{\text{mi}}{\text{in}} = 18\text{ miles}

Kara hikes 18 miles.

Check: 3 inches contains 3 \div \tfrac{1}{2} = 6 half-inches, and each half-inch is 3 miles: 6 \times 3 = 18 ✓

Tip: When a scale starts with a fraction of an inch, rewrite it as "1 inch = ___" first. Here \tfrac{1}{2} in = 3 mi becomes 1 in = 6 mi. Then add the map lengths and convert once.

1. Change the real length to inches so both lengths use the same unit: 58 \times 12 = 696\text{ in}
2. The scale 1 : 87 means model : real. Divide the real length by 87: 696 \div 87 = 8\text{ in}

The model locomotive is 8 inches long.

Check: 8 \times 87 = 696 inches, and 696 \div 12 = 58 feet ✓

Tip: A ratio scale like 1 : 87 has no units — it works only when both lengths are in the same unit. Convert first, then divide (real → model) or multiply (model → real).

1. Find the real side length. A square's area is side², so s = \sqrt{576} = 24\text{ ft}
2. 4 inches on the plan stands for 24 feet: \frac{24\text{ ft}}{4\text{ in}} = 6\text{ ft per inch}

The scale is 1 inch = 6 feet.

Check: The real patio is 4 \times 6 = 24 ft on a side, and 24 \times 24 = 576 ft² ✓

Tip: A scale compares lengths. If you are given an area, take the square root first to get back to a length, then divide by the drawing length.

1. Scale = real distance ÷ map distance (miles per inch): \frac{210\text{ mi}}{3.5\text{ in}}
2. Multiply top and bottom by 2 to clear the decimal: \frac{420}{7} = 60

The scale is 1 inch = 60 miles.

Check: 3.5 \times 60 = 210 miles ✓

Tip: To find a scale, divide the real distance by the map distance so the result is "miles per 1 inch." Clearing a decimal (×2 here) makes the division easy without a calculator.

1. Real length: 8 \times 30 = 240\text{ ft}
2. Real width: 5 \times 30 = 150\text{ ft}
3. Real area: 240 \times 150 = 36{,}000\text{ ft}^2

The park's area is 36,000 square feet.

Check: Using the area rule: drawing area = 8 \times 5 = 40 sq in, and area scales by 30^2 = 900: 40 \times 900 = 36{,}000 ✓

Tip: Never multiply a drawing area by the length scale (that gives 1,200). Either convert each side to real feet first, or multiply the drawing area by the square of the scale.

1. Real dimensions: 6 \times 25 = 150\text{ ft}, \quad 3.2 \times 25 = 80\text{ ft}
2. Real area: 150 \times 80 = 12{,}000\text{ ft}^2
3. Cost: 12{,}000 \times \$2.50 = \$30{,}000

Resurfacing will cost $30,000.

Check: The drawing's area is 6 \times 3.2 = 19.2 in², and each square inch stands for 25^2 = 625 ft²: 19.2 \times 625 = 12{,}000 ft² ✓

Tip: Convert the lengths to real units before you find the area. If you find the drawing's area first, multiply by the scale squared (625), not by 25.

1. Count how many 10-mile groups are in 65 miles: 65 \div 10 = 6.5
2. Each 10-mile group is \tfrac{3}{4} inch on the map: 6.5 \times \tfrac{3}{4} = \tfrac{13}{2} \times \tfrac{3}{4} = \tfrac{39}{8} = 4\tfrac{7}{8}\text{ in}

The towns are 4\tfrac{7}{8} inches apart on the map.

Check: 4\tfrac{7}{8} = 4.875 in. Converting back: 4.875 \div 0.75 = 6.5 groups, and 6.5 \times 10 = 65 miles ✓

Tip: Write the proportion with the same units on top: \frac{3/4\text{ in}}{10\text{ mi}} = \frac{x\text{ in}}{65\text{ mi}}, so x = 65 \times \tfrac{3}{4} \div 10. Ignoring the \tfrac{3}{4} (pretending the scale is 1 inch) is the most common slip.

1. When every length is multiplied by k, the area is multiplied by k^2: k^2 = 2.5^2 = 6.25
2. New area: 18 \times 6.25 = 112.5\text{ ft}^2

The new sign has an area of 112.5 square feet.

Check: Try a sign 3 ft by 6 ft (area 18 ft²). The new sign is 7.5 ft by 15 ft, and 7.5 \times 15 = 112.5 ft² ✓

Tip: Area is length × width, and both grow by the scale factor, so the area grows by the factor twice (k \times k). When no dimensions are given, picking easy sample numbers is a fast way to check.

1. Find the real length. Each \tfrac{1}{4} inch is 1 foot, so there are 4 feet per inch: 3\tfrac{1}{2} \times 4 = 14\text{ ft}
2. On the new blueprint each foot is \tfrac{3}{8} inch: 14 \times \tfrac{3}{8} = \tfrac{42}{8} = 5\tfrac{1}{4}\text{ in}

The bedroom is 5\tfrac{1}{4} inches long on the new blueprint.

Check: 5\tfrac{1}{4} \div \tfrac{3}{8} = \tfrac{21}{4} \times \tfrac{8}{3} = 14 ft ✓. The new scale gives more inches per foot, so the drawing should be bigger than 3\tfrac{1}{2} in — and it is.

Tip: Shortcut: new drawing = old drawing × (new inches per foot ÷ old inches per foot) = 3\tfrac{1}{2} \times \tfrac{3/8}{1/4} = 3\tfrac{1}{2} \times 1.5. Going through the real length is safer if you're unsure which way the ratio goes.

1. A unitless ratio needs both lengths in the same unit. Change 30 feet to inches: 30 \times 12 = 360\text{ in}
2. Write drawing : actual and divide both parts by 7.5: 7.5 : 360 = 1 : 48

The scale is 1 : 48.

Check: 7.5 \times 48 = 360 in, and 360 \div 12 = 30 ft ✓. (This is the common "\tfrac{1}{4} inch = 1 foot" architect's scale.)

Tip: Before writing a ratio scale, convert both lengths to one unit. Then divide both parts by the drawing length so the ratio starts with 1.

1. Real distance (from Map 1): 3.2 \times 25 = 80\text{ mi}
2. On Map 2, those 80 miles are shown as 5 inches. Miles per inch: \frac{80\text{ mi}}{5\text{ in}} = 16\text{ mi per in}

Map 2's scale is 1 inch = 16 miles.

Check: 5 \times 16 = 80 mi, the same real distance as on Map 1 ✓ Map 2 shows the lakes farther apart (5 in vs 3.2 in), so each inch must cover fewer miles than on Map 1 — and 16 < 25 ✓

Tip: When two scales describe the same real object, the real distance is the bridge between them. Find it first, then divide by the new map length.

1. Difference on the model: 9 - 1.75 = 7.25\text{ in}
2. Scale up to real size (real = 48 × model): 7.25 \times 48 = 348\text{ in}
3. Convert to feet: 348 \div 12 = 29\text{ ft}

The real house is 29 feet taller than the front door.

Check: Real house: 9 \times 48 = 432 in = 36 ft. Real door: 1.75 \times 48 = 84 in = 7 ft. 36 - 7 = 29 ft ✓

Tip: A ratio scale is unitless: the real length is 48 times the model length in the same unit. Do the scale step and the unit conversion as two separate steps, and read the question twice — here it asks for a difference.

1. Small triangle: the woman's height is 5 ft and its base is her shadow, 8 ft.
2. Large triangle: the lamp's height is h, and its base runs from the lamp to the tip of the shadow: 12 + 8 = 20\text{ ft}
3. Proportion (height : base): \frac{5}{8} = \frac{h}{20}
4. Solve: h = 20 \times \frac{5}{8} = \frac{100}{8} = 12.5\text{ ft}

The lamp post is 12.5 feet tall.

Check: \frac{12.5}{20} = 0.625 and \frac{5}{8} = 0.625 ✓

Tip: Sketch the picture. Both triangles share the shadow's tip, so the big triangle's base is the whole distance from the lamp to the tip (12 + 8), not just 12. Matching the wrong sides is the most common error in shadow problems.