ANTMACK215 × WEALTHWISEGUYS
WealthWiseGuys Practical Mathematics

Chapter 27

Ratios, Geometry, Measurement, and Business Applications

From recipe parts to floor plans and container capacity

A business buys space, materials, and quantities as well as products. Use ratios and measurement to turn hypothetical plans into workable estimates, with clear units and sensible rounding.

27.1

Ratios, Rates, and Scale Drawings

Learning objectives

After this section you will be able to:

  1. Simplify ratios while preserving order.
  2. Use unit rates and proportions in business decisions.
  3. Translate between drawing lengths and actual lengths.

Real-life situation

Ratios, Rates, and Scale Drawings in practice

A hypothetical food truck near a Philadelphia recreation center plans a stand layout and a drink batch. All quantities and prices are invented. Round final money to cents and preserve units throughout.

Drawing length (cm)Actual length (m)
12
36
48
Figure 27.1 Ratios, Rates, and Scale Drawings (hypothetical data)

The concept

A ratio compares two quantities in a stated order. A syrup-to-water ratio of 2:5 means two parts syrup for every five parts water, not two parts syrup out of five total. There are seven total parts. Labeling the quantities prevents a surprisingly common mixing error.

A rate compares quantities with different units, such as dollars per kilogram or meals per hour. Divide to find a unit rate, then compare like units. A lower package price can still have a higher cost per unit if the package is smaller.

A proportion states that two ratios are equal. Multiplying or dividing both quantities by the same positive factor preserves a mixture or scale. Cross products provide another method, but units should still tell you what the unknown represents.

A scale drawing replaces actual distances with proportional drawing distances. If 1 cm represents 2 m, convert each measured length using that relation. Area changes with the square of the length factor, so a drawing-area conversion cannot use the length multiplier just once.

Ratios, Rates, and Scale Drawings: solution map

  1. 1

    Identify the question

    Make 21 liters of drink using syrup:water = 2:5.

  2. 2

    Calculation 1

    Total parts = 2 + 5 = 7.

  3. 3

    Calculation 2

    Each part = 21 ÷ 7 = 3 liters.

  4. 4

    Calculation 3

    Syrup = 2 × 3 = 6 liters.

  5. 5

    Calculation 4

    Water = 5 × 3 = 15 liters; 6 + 15 = 21.

  6. 6

    Interpret and check

    Use 6 liters syrup and 15 liters water.

Figure 27.1.V Ratios, Rates, and Scale Drawings: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • cost

    The amount paid to make or buy an item before it is sold.

  • markup

    The amount added to a cost to set a selling price; the markup rate uses cost as its base.

  • profit margin

    Profit shown as a percent of revenue; a 10% margin means $10 of profit per $100 of sales.

Key terms

Ratio
An ordered comparison of two quantities by division.
Unit rate
A rate with a denominator of one unit.
Proportion
An equation asserting that two ratios are equal.
Scale
A fixed relationship between drawing and actual measurements.
Part-to-whole ratio
A comparison of a component with the entire amount.

Formulas & rules

  • Unit rate

    Unit rate = total quantity ÷ number of units

  • Proportion

    a ÷ b = c ÷ d means a × d = b × c

  • Scale conversion

    Actual length = drawing length × actual units per drawing unit

Worked examples

Example 1A drink batch

Make 21 liters of drink using syrup:water = 2:5.

Solution

  1. 1Total parts = 2 + 5 = 7.
  2. 2Each part = 21 ÷ 7 = 3 liters.
  3. 3Syrup = 2 × 3 = 6 liters.
  4. 4Water = 5 × 3 = 15 liters; 6 + 15 = 21.

Answer: Use 6 liters syrup and 15 liters water.

Example 2A booth plan

A rectangular booth is 3 cm by 4 cm on a drawing with 1 cm representing 2 m. Find actual dimensions and area.

Solution

  1. 1Actual short side = 3 × 2 = 6 m.
  2. 2Actual long side = 4 × 2 = 8 m.
  3. 3Actual area = 6 × 8 = 48 m².
  4. 4Drawing area = 12 cm²; area multiplier = 2² = 4 m² per cm².

Answer: The booth is 6 m by 8 m, with area 48 m².

Practice exercises 27.1

  1. Simplify a 12:18 ratio of cups to lids.
  2. Compare 3 kg for $12 with 5 kg for $18.
  3. A recipe uses 4 cups flour for 10 servings. How much for 25 servings?
  4. At 1 cm to 3 m, what actual length does 7 cm represent?
  5. A 1:4 concentrate-to-water mixture totals 15 liters. Find concentrate.

Recap quiz 27.1

1. Simplify 8:12.
2. Cost per notebook for 6 costing $15?
3. A length scale factor of 3 gives what area factor?

Section 27.1 recap

  • Keep ratio order and units explicit.
  • Scale lengths once and areas by the square of the length factor.
Answer key with explanations: Section 27.1

Practice exercises

  1. Divide both terms by 6: 12:18 = 2:3; order remains cups:lids.
  2. 12 ÷ 3 = $4.00/kg; 18 ÷ 5 = $3.60/kg. The 5 kg package costs $0.40 less per kg.
  3. Scale factor = 25 ÷ 10 = 2.5; flour = 4 × 2.5 = 10 cups.
  4. 7 × 3 = 21 m.
  5. There are 1 + 4 = 5 parts. Each is 15 ÷ 5 = 3 liters, so concentrate is 3 liters, not 15 ÷ 4.

Recap quiz

  1. 2:3: Divide both terms by 4.
  2. $2.50: 15 ÷ 6 = $2.50 per notebook.
  3. 9: Both length and width multiply by 3, so area multiplies by 3² = 9.
27.2

Area, Perimeter, and Material Costs

Learning objectives

After this section you will be able to:

  1. Distinguish boundary length from surface area.
  2. Calculate rectangular, triangular, and circular measurements.
  3. Estimate material purchases with waste and package rounding.

Real-life situation

Area, Perimeter, and Material Costs in practice

A hypothetical student vendor needs floor covering and boundary tape for a 6 m by 4 m rectangular booth. Prices exclude tax; no current tax rules are implied. Round money to cents and round package counts upward.

QuantityAmount
Length6 m
Width4 m
Area24 m²
Perimeter20 m
Figure 27.2 Area, Perimeter, and Material Costs (hypothetical data)

The concept

Perimeter measures distance around a boundary, so it uses length units. Area measures the surface enclosed, so it uses square units. A booth may need tape around its edge and mats across its floor; those are different purchasing problems.

Rectangle area is length times width, while its perimeter counts two lengths and two widths. A triangle occupies half the area of a rectangle with the same base and perpendicular height. The slanted side is not the height unless it is perpendicular to the base.

For circles, radius runs from center to edge and diameter is twice the radius. Circumference measures the curved boundary; area uses radius squared. Use the calculator value of π unless a problem specifies an approximation.

An estimate should allow for cuts, breakage, and packaging. Add a stated waste allowance to the required area, divide by coverage per package, then round up to whole packages. Waste assumptions are planning choices, not universal construction standards; also account for openings and overlap when relevant.

Area, Perimeter, and Material Costs: solution map

  1. 1

    Identify the question

    Cover a 6 m by 4 m booth with 10% waste. Each pack covers 5 m² and costs $18.

  2. 2

    Calculation 1

    Area = 6 × 4 = 24 m².

  3. 3

    Calculation 2

    Required material = 24 × 1.10 = 26.4 m².

  4. 4

    Calculation 3

    Packs = 26.4 ÷ 5 = 5.28; buy 6.

  5. 5

    Calculation 4

    Cost = 6 × $18 = $108.00.

  6. 6

    Interpret and check

    Buy 6 packs for $108.00 before any tax.

Figure 27.2.V Area, Perimeter, and Material Costs: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • inventory

    The stock of goods a business holds ready to sell.

  • overhead

    Ongoing business costs such as rent, utilities, and insurance that do not change with each sale.

  • budget

    A plan that shows expected income and how it will be spent.

Key terms

Perimeter
Total distance around a polygon.
Area
The amount of enclosed surface in square units.
Radius
Distance from the center of a circle to its edge.
Perpendicular height
A height meeting its base at a right angle.
Waste allowance
Extra material budgeted for unusable cuts or damage.

Formulas & rules

  • Rectangle

    A = l × w; P = 2(l + w)

  • Triangle

    A = b × h ÷ 2

  • Circle

    A = πr²; C = 2πr

  • Material planning

    Packages = round up [area × (1 + waste rate) ÷ coverage per package]

Worked examples

Example 1Covering the booth

Cover a 6 m by 4 m booth with 10% waste. Each pack covers 5 m² and costs $18.

Solution

  1. 1Area = 6 × 4 = 24 m².
  2. 2Required material = 24 × 1.10 = 26.4 m².
  3. 3Packs = 26.4 ÷ 5 = 5.28; buy 6.
  4. 4Cost = 6 × $18 = $108.00.

Answer: Buy 6 packs for $108.00 before any tax.

Example 2A round sign

Find the area and edge length of a circular sign of radius 0.5 m.

Solution

  1. 1Diameter = 2 × 0.5 = 1 m.
  2. 2Area = π × 0.5² = 0.25π ≈ 0.7854 m².
  3. 3Circumference = 2π × 0.5 = π ≈ 3.1416 m.
  4. 4Round measurements to two decimals: 0.79 m² and 3.14 m.

Answer: Area about 0.79 m²; edge about 3.14 m.

Practice exercises 27.2

  1. Find area and perimeter of a 9 m by 3 m rectangle.
  2. Find triangle area with base 8 cm and perpendicular height 5 cm.
  3. How many 4 m² packs cover 30 m² with no waste allowance?
  4. Tape costs $1.50 per meter. Tape all edges of a 6 m by 4 m booth.
  5. A square side doubles from 2 m to 4 m. Compare areas.

Recap quiz 27.2

1. Which units measure floor area?
2. Area of a triangle with base 6 and height 4?
3. Need 10.1 packages; how many must be bought?

Section 27.2 recap

  • Match the formula and units to the material being purchased.
  • Apply waste before rounding package counts upward.
Answer key with explanations: Section 27.2

Practice exercises

  1. Area = 9 × 3 = 27 m²; perimeter = 2(9 + 3) = 24 m.
  2. 8 × 5 ÷ 2 = 20 cm².
  3. 30 ÷ 4 = 7.5 packs; round up to 8 packs, covering 32 m².
  4. Perimeter = 2(6 + 4) = 20 m; cost = 20 × 1.50 = $30.00.
  5. Original area = 2² = 4 m²; new area = 4² = 16 m². Ratio = 16 ÷ 4 = 4, not 2.

Recap quiz

  1. Square meters: Area counts square units.
  2. 12: 6 × 4 ÷ 2 = 12 square units.
  3. 11: Whole packages require rounding up to 11.
27.3

Volume, Conversions, and Capacity

Learning objectives

After this section you will be able to:

  1. Calculate box and cylinder volumes.
  2. Convert units before combining measurements.
  3. Use capacity and density to plan business quantities.

Real-life situation

Volume, Conversions, and Capacity in practice

A hypothetical food truck uses a rectangular water tank with inside dimensions 50 cm by 40 cm by 30 cm. Capacity calculations ignore wall thickness because dimensions are internal. Money is rounded to cents; physical limits are rounded conservatively.

MeasurementValue
Length50 cm
Width40 cm
Height30 cm
Capacity60 L
Figure 27.3 Volume, Conversions, and Capacity (hypothetical data)

The concept

Volume measures three-dimensional space in cubic units. A rectangular container has volume length times width times height. A cylinder has a circular base area multiplied by its perpendicular height. Use internal dimensions to estimate usable capacity.

Convert all dimensions to compatible units before multiplying. Since a meter contains 100 centimeters, a cubic meter contains 100³ cubic centimeters, not just 100. A liter equals 1,000 cm³, and a milliliter equals 1 cm³.

Capacity is the amount a container can hold. A usable-fill percentage may reserve headspace for expansion or safe handling. Once the usable amount is known, divide by serving size and round down to count complete servings; rounding up would promise product you do not have.

Volume and mass are not interchangeable. Density connects them through mass per unit volume. A delivery weight estimate must include both product mass and container mass, and real loading plans require verified equipment limits rather than a textbook estimate.

Volume, Conversions, and Capacity: solution map

  1. 1

    Identify the question

    Find tank capacity at 80% fill for internal dimensions 50 cm by 40 cm by 30 cm, and count 250 mL servings.

  2. 2

    Calculation 1

    Volume = 50 × 40 × 30 = 60,000 cm³.

  3. 3

    Calculation 2

    Capacity = 60,000 ÷ 1,000 = 60 L.

  4. 4

    Calculation 3

    Usable volume = 60 × 0.80 = 48 L = 48,000 mL.

  5. 5

    Calculation 4

    Servings = 48,000 ÷ 250 = 192.

  6. 6

    Interpret and check

    Usable capacity 48 L, enough for 192 full servings.

Figure 27.3.V Volume, Conversions, and Capacity: solution map. Hypothetical teaching data.

Money Words

Plain-language words you will use in this lesson — learn them before we reach the formulas.

  • expense

    Money a business spends to run, such as rent, supplies, or wages.

  • invoice

    A document listing goods or services sold and the amount owed.

  • break-even point

    The sales level where revenue exactly equals total cost, so there is no profit and no loss.

Key terms

Volume
Space occupied or enclosed, expressed in cubic units.
Capacity
The amount a container can hold.
Liter
A volume unit equal to 1,000 cm³.
Density
Mass per unit volume.
Tare mass
The mass of an empty container.

Formulas & rules

  • Rectangular volume

    V = l × w × h

  • Cylinder volume

    V = πr²h

  • Mass from density

    Mass = density × volume

  • Serving count

    Complete servings = round down (usable volume ÷ serving volume)

Worked examples

Example 1Tank capacity

Find tank capacity at 80% fill for internal dimensions 50 cm by 40 cm by 30 cm, and count 250 mL servings.

Solution

  1. 1Volume = 50 × 40 × 30 = 60,000 cm³.
  2. 2Capacity = 60,000 ÷ 1,000 = 60 L.
  3. 3Usable volume = 60 × 0.80 = 48 L = 48,000 mL.
  4. 4Servings = 48,000 ÷ 250 = 192.

Answer: Usable capacity 48 L, enough for 192 full servings.

Example 2Cylindrical container

A cylindrical container has inside radius 10 cm and height 30 cm. Find liters to two decimals.

Solution

  1. 1Base area = π × 10² = 100π cm².
  2. 2Volume = 100π × 30 = 3,000π cm³.
  3. 3Liters = 3,000π ÷ 1,000 = 3π L.
  4. 4Round 3π ≈ 9.42478 L to 9.42 L.

Answer: Capacity is about 9.42 L.

Practice exercises 27.3

  1. Convert 2.5 L to milliliters.
  2. Find liters in a box 20 cm by 30 cm by 40 cm.
  3. How many complete 300 mL servings come from 5 L?
  4. A liquid has density 1.2 kg/L. Find the loaded mass of 10 L in a 2 kg container.
  5. A box is 0.5 m by 20 cm by 30 cm. Find liters.

Recap quiz 27.3

1. 1 L equals how many cm³?
2. Box volume uses which calculation?
3. 8 L available, each serving 0.3 L. Full servings?

Section 27.3 recap

  • Convert dimensions first, then calculate volume.
  • Round servings down and include container mass in load estimates.
Answer key with explanations: Section 27.3

Practice exercises

  1. 2.5 × 1,000 = 2,500 mL.
  2. 20 × 30 × 40 = 24,000 cm³; divide by 1,000 to get 24 L.
  3. 5 L = 5,000 mL; 5,000 ÷ 300 ≈ 16.67. Round down to 16; 200 mL remains.
  4. Product mass = 1.2 × 10 = 12 kg; add tare 2 kg to obtain 14 kg.
  5. Convert 0.5 m = 50 cm. Volume = 50 × 20 × 30 = 30,000 cm³ = 30 L.

Recap quiz

  1. 1,000: A liter contains 1,000 cubic centimeters.
  2. l × w × h: Three dimensions multiply to cubic units.
  3. 26: 8 ÷ 0.3 ≈ 26.67, so only 26 complete servings.

Chapter 27 Organizer and Review

Chapter Organizer

Every section at a glance: key formulas, key terms, and the main idea.

  • 27.1 Ratios, Rates, and Scale Drawings

    Key formulas

    • Unit rate: Unit rate = total quantity ÷ number of units
    • Proportion: a ÷ b = c ÷ d means a × d = b × c
    • Scale conversion: Actual length = drawing length × actual units per drawing unit

    Key terms

    Ratio, Unit rate, Proportion, Scale, Part-to-whole ratio

    Remember

    Keep ratio order and units explicit.

  • 27.2 Area, Perimeter, and Material Costs

    Key formulas

    • Rectangle: A = l × w; P = 2(l + w)
    • Triangle: A = b × h ÷ 2
    • Circle: A = πr²; C = 2πr
    • Material planning: Packages = round up [area × (1 + waste rate) ÷ coverage per package]

    Key terms

    Perimeter, Area, Radius, Perpendicular height, Waste allowance

    Remember

    Match the formula and units to the material being purchased.

  • 27.3 Volume, Conversions, and Capacity

    Key formulas

    • Rectangular volume: V = l × w × h
    • Cylinder volume: V = πr²h
    • Mass from density: Mass = density × volume
    • Serving count: Complete servings = round down (usable volume ÷ serving volume)

    Key terms

    Volume, Capacity, Liter, Density, Tare mass

    Remember

    Convert dimensions first, then calculate volume.

Cumulative Review

Mixed practice from this chapter and the two chapters before it.

  1. Section 27.1

    A 1:4 concentrate-to-water mixture totals 15 liters. Find concentrate.

    Show answer

    There are 1 + 4 = 5 parts. Each is 15 ÷ 5 = 3 liters, so concentrate is 3 liters, not 15 ÷ 4.

  2. Section 27.2

    A square side doubles from 2 m to 4 m. Compare areas.

    Show answer

    Original area = 2² = 4 m²; new area = 4² = 16 m². Ratio = 16 ÷ 4 = 4, not 2.

  3. Section 27.3

    A box is 0.5 m by 20 cm by 30 cm. Find liters.

    Show answer

    Convert 0.5 m = 50 cm. Volume = 50 × 20 × 30 = 30,000 cm³ = 30 L.

  4. Review of Section 25.3

    Find the vertex and maximum of P(q) = −q² + 12q − 20 for q ≥ 0.

    Show answer

    qᵥ = −12 ÷ (−2) = 6. P(6) = −36 + 72 − 20 = 16. Since a < 0, the maximum is 16 at q = 6.

  5. Review of Section 26.3

    Find $500 after 2 years at 4% compounded annually.

    Show answer

    500 × 1.04² = $540.80.

All prices, stores, people, and accounts in this chapter are hypothetical and for teaching only. Product references are generic categories; no brand endorses or is endorsed by this textbook.

Chapter 26Chapter 28