Chapter 1
Number and Quantity
Course: Core Academic Skills For Educators Combined Test 5752
Subject: Core Academic Skills Educators Mathematics 5733
Chapter: Number And Quantity
1. Understanding Ratios
A ratio expresses a relationship between two quantities by division. It tells how many times one quantity is compared to another. Ratios can be written in several equivalent forms: using a colon (a : b), as a fraction (a/b), or in words (“a to b”).
There are two main types of ratios:
Part-to-part ratio compares one part of a whole to another part. For example, in a fruit basket containing 15 apples and 10 oranges, the ratio of apples to oranges is 15:10, which can be simplified to 3:2.
Part-to-whole ratio compares one part of a whole to the total quantity. In the same basket, the ratio of apples to total fruit is 15:25, which simplifies to 3:5.
Ratios can be scaled up or down by multiplying or dividing both terms by the same non-zero number. This scaling preserves equivalence. For example, the ratio 4:5 is equivalent to 8:10 because both terms are multiplied by 2.
In real life, ratios appear in recipes, mixing paints, construction measurements, financial comparisons, and statistical analysis.
2. Understanding Proportions
A proportion is a statement that two ratios are equal. In mathematical form:
a/b = c/d
This equality is valid if the cross-products are equal, meaning a × d = b × c.
Proportions are used to compare relationships, especially when one term is unknown. To solve for the unknown term, cross-multiplication is applied, followed by division to isolate the variable.
Example: If 3/4 = x/12, then 3 × 12 = 4 × x, leading to 36 = 4x, and x = 9.
In daily life, proportions are used when scaling recipes, enlarging images, converting measurements, determining exchange rates, or calculating probabilities.
3. Problem-Solving Process for Ratios and Proportions
Identify the quantities being compared and ensure they are in the same units.
Represent the relationship as a ratio.
If comparing equivalent situations, set up a proportion.
Use cross-multiplication to solve for the unknown.
Check the solution for reasonableness by substituting it back into the original relationship.
Care must be taken to keep the order of terms consistent; the first quantity in one ratio must correspond to the first quantity in the other ratio.
4. Common Applications
a) Unit Rates
A unit rate expresses a comparison to one unit of another quantity. For example, if a vehicle travels 300 miles in 5 hours, the unit rate (speed) is 300 ÷ 5 = 60 miles per hour. Unit rates are seen in grocery price tags (cost per pound), wages (dollars per hour), and travel (miles per gallon).
b) Scale Drawings and Maps
Ratios represent scale in diagrams or maps. For example, if a map scale is 1 inch : 20 miles, a measured distance of 3.5 inches corresponds to 3.5 × 20 = 70 miles in reality. This is crucial in architecture, engineering, and geography.
c) Mixture Problems
Ratios are applied to combine substances in specific proportions. For example, a cleaning solution with a 4:1 water-to-bleach ratio means that for 12 cups of water, bleach needed is calculated as 4/1 = 12/x, which gives x = 3 cups. This concept is also used in fuel blending, medication preparation, and material mixing.
d) Percent Problems
Percentages are ratios where the denominator is 100. For example, 18 is 60 percent of what number? Set up the proportion 60/100 = 18/x, which leads to x = 30. Percentages are used in discounts, tax calculations, interest rates, and statistical analysis.
e) Conversions
Changing from one unit to another often involves proportional reasoning. For example, converting 15 inches to centimeters given 1 inch = 2.54 cm means setting 1/2.54 = 15/x, giving x = 38.1 cm. This is common in scientific measurements, travel, and manufacturing.
5. Advanced and Technical Considerations
Dimensional analysis is a systematic method of applying proportional reasoning to conversions. Units are treated like variables and can be canceled to ensure the result is in the correct units.
Example: (120 miles / 3 hours) × (1 hour / 60 minutes) = 40 miles per hour.
Inverse proportions occur when one quantity increases as another decreases, such that their product remains constant. For example, if speed increases for a fixed distance, travel time decreases proportionally.
Ratios may also be expressed as fractions, decimals, or percentages depending on the context. For instance, a probability ratio of 2:3 can be expressed as a fraction 2/5 or a decimal 0.4.
6. Common Mistakes to Avoid
Mixing up the order of terms in a ratio, leading to incorrect proportional relationships.
Forgetting to convert all quantities into consistent units before setting up ratios.
Using proportions in situations where the relationship is not truly proportional, such as comparing variables with non-linear relationships.
Rounding too early, which can cause compounding errors in multi-step problems.
7. Real-Life Examples
Education: Determining the student-teacher ratio in a school (e.g., 1 teacher per 25 students).
Medicine: Calculating medication dosage by body weight (e.g., milligrams per kilogram).
Construction: Mixing concrete in a cement : sand : gravel ratio, such as 1:2:3.
Finance: Determining exchange rates, such as 1 US dollar = 0.92 euros, and scaling for large transactions.
Retail: Determining discount prices based on percentage reductions from original price.
Sports: Calculating batting averages or win-loss ratios.