Ratio Conversion
1. Understanding Ratios
A ratio is a comparison of two or more quantities using division. Ratios can be written in several forms:
- Using a colon: 3:4
- As a fraction: \(\frac{3}{4}\)
- Using the word "to": 3 to 4
All three forms represent the same relationship. When we work with ratios mathematically, the fraction form is often most useful because we can apply our knowledge of equivalent fractions and proportions.
Key Rule: A ratio compares parts to parts, or parts to whole. The order matters! A ratio of 3:4 is different from 4:3.
How the HSPT tests this: Questions often require you to identify which quantities are being compared and in what order. Common traps include reversing the ratio or confusing part-to-part ratios with part-to-whole ratios.
1.1 Part-to-Part vs. Part-to-Whole Ratios
Understanding the difference between these two types of ratios is essential:
- Part-to-Part: Compares one part of a group to another part. Example: In a class with 12 boys and 15 girls, the ratio of boys to girls is 12:15 or 4:5.
- Part-to-Whole: Compares one part to the entire group. Example: The ratio of boys to total students is 12:27 or 4:9.
When converting between these, remember that the whole equals the sum of all parts.
Example: If a mixture contains ingredients in the ratio 2:3, and you know there are 2 parts of the first ingredient, there are 3 parts of the second ingredient, making 5 parts total.
2. Converting Ratios to Fractions
Converting a ratio to a fraction allows us to perform calculations more easily. The key is understanding what the numerator and denominator represent.
2.1 Part-to-Part Ratios as Fractions
A part-to-part ratio like 3:5 can be written as the fraction \(\frac{3}{5}\), which tells us "for every 3 of the first quantity, there are 5 of the second quantity."
This fraction represents a relationship, not necessarily actual quantities. If we have 3 red marbles and 5 blue marbles, the ratio is 3:5. But if we have 6 red and 10 blue, the ratio is still 3:5 (simplified).
2.2 Converting Part-to-Part to Part-to-Whole
When you have a part-to-part ratio and need to find what fraction each part represents of the whole:
Method:
1. Add the parts to find the total
2. Write each part as a fraction with the total as the denominator
Example: Ratio 2:3
Total parts = 2 + 3 = 5
First quantity is \(\frac{2}{5}\) of the whole
Second quantity is \(\frac{3}{5}\) of the whole
How the HSPT tests this: You might be given a ratio and a total amount, then asked what portion goes to each part. Students often forget to add the ratio parts together first, leading to incorrect denominators.
Example: In a bag containing red and blue marbles in the ratio 3:7, what fraction of the marbles are red?
- \(\frac{3}{7}\)
- \(\frac{3}{10}\)
- \(\frac{7}{10}\)
- \(\frac{1}{3}\)
Correct Answer: (B)
Solution:
The ratio of red to blue is 3:7
Total parts = 3 + 7 = 10
Red marbles represent 3 parts out of 10 total parts
Fraction of red marbles = \(\frac{3}{10}\)
Why each wrong answer is a trap:
(A) represents the part-to-part ratio (red to blue) instead of part-to-whole
(C) represents the fraction of blue marbles, not red
(D) results from incorrectly dividing 3 by (3 + 7) but making an arithmetic error
3. Converting Ratios to Percentages
Converting ratios to percentages is a two-step process: first convert to a fraction, then convert the fraction to a percentage.
3.1 Basic Conversion Method
Steps to convert a ratio to a percentage:
1. Convert the ratio to a part-to-whole fraction
2. Multiply the fraction by 100
3. Add the percent symbol (%)
Example: If boys and girls are in the ratio 2:3, what percentage of the students are boys?
- Total parts = 2 + 3 = 5
- Boys represent \(\frac{2}{5}\) of the total
- \(\frac{2}{5} \times 100 = \frac{200}{5} = 40\%\)
How the HSPT tests this: Questions may ask for the percentage of one part, then offer trap answers showing the percentage of the other part, or the part-to-part ratio treated as a percentage directly.
3.2 Working Backwards from Percentages
Sometimes you'll need to convert a percentage back to a ratio. This requires understanding that percentages are always out of 100.
Example: If 40% of students are boys, what is the ratio of boys to girls?
- 40% are boys means 60% are girls
- Ratio of boys to girls = 40:60 = 2:3 (simplified)
Example: A store marks up items by 25%. What is the ratio of the cost price to the selling price?
- 4:5
- 5:4
- 1:4
- 3:4
Correct Answer: (A)
Solution:
Let the cost price be 100 (using 100 makes percentage calculations easy)
Markup is 25% of 100 = 25
Selling price = 100 + 25 = 125
Ratio of cost to selling price = 100:125
Simplify by dividing both by 25: 4:5
Why each wrong answer is a trap:
(B) reverses the ratio (selling price to cost price instead of cost to selling)
(C) results from incorrectly thinking 25% means 1 part in 4, confusing percentage with ratio
(D) results from incorrectly subtracting 25% from 100% and treating as selling price
4. Scaling Ratios
Ratios describe relationships that remain constant even when quantities change. Scaling means finding equivalent ratios by multiplying or dividing all parts by the same number.
4.1 Equivalent Ratios
Just like equivalent fractions, equivalent ratios represent the same relationship:
- 2:3 = 4:6 = 6:9 = 8:12 (multiply both parts by the same number)
- 12:18 = 6:9 = 2:3 (divide both parts by the same number)
Key Rule: To maintain the same ratio, you must multiply or divide ALL parts by the SAME number.
How the HSPT tests this: You might be given actual quantities and asked to find a ratio, or given a ratio and one quantity and asked to find the other. Trap answers often result from adding instead of multiplying, or changing only one part of the ratio.
4.2 Finding Missing Values in Proportions
When ratios are equal, we have a proportion. This is useful for finding unknown values.
Method using cross-multiplication:
If \(\frac{a}{b} = \frac{c}{d}\), then \(a \times d = b \times c\)
Example: If 3 pencils cost $2, how much do 12 pencils cost?
- Set up the proportion: \(\frac{3}{2} = \frac{12}{x}\)
- Cross-multiply: \(3x = 2 \times 12 = 24\)
- Solve: \(x = 8\)
Example: A recipe calls for flour and sugar in the ratio 5:2. If you use 15 cups of flour, how many cups of sugar do you need?
- 4 cups
- 6 cups
- 8 cups
- 10 cups
Correct Answer: (B)
Solution:
Original ratio is flour:sugar = 5:2
We have 15 cups of flour, which is 5 × 3
To maintain the ratio, multiply both parts by 3
New ratio: 15:6
Sugar needed = 6 cups
Alternative method using proportion:
\(\frac{5}{2} = \frac{15}{x}\)
Cross-multiply: \(5x = 30\)
\(x = 6\)
Why each wrong answer is a trap:
(A) results from adding 2 to the original ratio of 2, not scaling proportionally
(C) results from incorrectly doubling the sugar when flour is tripled
(D) results from adding 5 + 2 = 7, then adding 3 more
5. Converting Between Ratio Forms
The HSPT often requires you to move flexibly between different ways of expressing the same relationship: ratios, fractions, decimals, and percentages.
5.1 Ratio to Decimal
Convert the ratio to a fraction, then divide:
- Ratio 3:5 → Fraction \(\frac{3}{5}\) → Decimal 0.6
- This represents the relationship of the first quantity to the second
For part-to-whole conversions, remember to find the total first.
5.2 Multi-Part Ratios
Some ratios compare three or more quantities, like 2:3:5. The same principles apply:
- Total parts = 2 + 3 + 5 = 10
- First quantity is \(\frac{2}{10} = \frac{1}{5}\) of the whole
- Second quantity is \(\frac{3}{10}\) of the whole
- Third quantity is \(\frac{5}{10} = \frac{1}{2}\) of the whole
How the HSPT tests this: Multi-part ratios appear in mixture problems, sharing problems, and geometric problems involving angles or side lengths. Students often make errors by forgetting to include all parts when finding the total.
Example: Three siblings share an inheritance in the ratio 4:5:6. If the total inheritance is $45,000, how much does the sibling with the largest share receive?
- $12,000
- $15,000
- $18,000
- $27,000
Correct Answer: (C)
Solution:
The ratio is 4:5:6
Total parts = 4 + 5 + 6 = 15
The largest share is 6 parts
Value of one part = $45,000 ÷ 15 = $3,000
Largest share = 6 × $3,000 = $18,000
Why each wrong answer is a trap:
(A) represents 4 parts (the smallest share), not the largest
(B) represents 5 parts (the middle share)
(D) results from incorrectly treating 6 as the denominator and calculating \(\frac{6}{10}\) of $45,000
6. Application Problems
Ratio conversion appears in many real-world contexts. The key is identifying what ratio is being described and what conversion is needed.
6.1 Mixture and Concentration Problems
These problems involve combining substances in given ratios or finding the concentration of mixtures.
Common scenario: A paint mixture uses blue and yellow in ratio 3:2. This means for every 3 parts blue, there are 2 parts yellow, making 5 parts total.
6.2 Scale and Map Problems
Scale ratios tell us the relationship between measurements on a drawing or map and actual measurements.
Example: A map scale of 1:50,000 means 1 cm on the map represents 50,000 cm in reality.
To convert: multiply the map measurement by the scale factor.
How the HSPT tests this: You may need to convert units as well as apply the ratio. Trap answers often result from forgetting unit conversions or applying the scale factor incorrectly.
Example: On a map with scale 1:40,000, two cities are 7 cm apart. What is the actual distance between the cities in kilometers?
- 0.28 km
- 2.8 km
- 28 km
- 280 km
Correct Answer: (B)
Solution:
Scale 1:40,000 means 1 cm on map = 40,000 cm in reality
Map distance = 7 cm
Actual distance = 7 × 40,000 = 280,000 cm
Convert to km: 280,000 cm = 2,800 m = 2.8 km
(Remember: 100 cm = 1 m, and 1,000 m = 1 km)
Why each wrong answer is a trap:
(A) results from dividing 7 by 40,000 instead of multiplying, then converting
(C) results from correctly calculating 280,000 cm but converting as if 10,000 cm = 1 km
(D) results from forgetting to convert from cm to km entirely
6.3 Speed, Distance, and Time
Speed is a ratio of distance to time. Converting between different units requires ratio conversion skills.
Example: A car travels at 60 miles per hour. This is a ratio of 60 miles : 1 hour, or \(\frac{60 \text{ miles}}{1 \text{ hour}}\).
To find distance in 3 hours: \(60 \times 3 = 180\) miles.
7. Practice Questions
1. In a class, the ratio of students who walk to school to those who ride the bus is 3:7. If there are 30 students in the class, how many ride the bus?
- 9
- 12
- 18
- 21
2. A paint mixture requires red and white paint in the ratio 2:9. What percentage of the mixture is red paint?
- 18%
- 20%
- 22%
- 25%
3. The angles of a triangle are in the ratio 2:3:4. What is the measure of the largest angle in degrees?
- 40°
- 60°
- 80°
- 90°
4. If 5 notebooks cost $8, how much would 15 notebooks cost?
- $13
- $20
- $24
- $40
5. A recipe for 4 servings calls for 3 cups of rice. How many cups of rice are needed for 10 servings?
- 6 cups
- 7 cups
- 7.5 cups
- 8 cups
6. In a school, the ratio of boys to girls is 5:6. If there are 120 boys, how many girls are there?
- 100
- 125
- 144
- 150
7. A map has a scale of 1:25,000. If two points are 4 cm apart on the map, what is the actual distance between them in meters?
- 100 m
- 1,000 m
- 10,000 m
- 100,000 m
8. Three friends split a restaurant bill in the ratio 3:4:5. If the total bill is $72, how much does the person with the smallest share pay?
- $15
- $18
- $24
- $30
9. A solution contains water and concentrate in the ratio 7:2. If there are 63 liters of water, how many liters of concentrate are there?
- 9 liters
- 14 liters
- 18 liters
- 21 liters
10. A store reduces prices by 20%. What is the ratio of the new price to the original price?
- 1:5
- 4:5
- 5:4
- 8:10
Answer Key
1. (D) | 2. (A) | 3. (C) | 4. (C) | 5. (C) | 6. (C) | 7. (B) | 8. (B) | 9. (C) | 10. (B)
