**Ratio and Proportion**

**Ratio is a way of comparing two quantities. **

**It tells you how much of one thing there is in comparison to another thing.**

**Example 1:**

**if you have 2 red marbles and 3 blue marbles, ****the ratio of red marbles to blue marbles is 2:3. **

**This means that for every 2 red marbles, there are 3 blue marbles.**

**Example 2:**

**If a recipe calls for 2 cups of flour and 1 cup of sugar, ****the ratio of flour to sugar is 2:1. **

**This means that for every 2 cups of flour, ****there is 1 cup of sugar.**

**Example 3:**

**A basketball team has won 15 games and lost 5 games. ****The ratio of wins to losses is 15:5**

**which can be simplified to 3:1. ****To simplify ratios, ****you can divide both sides of the ratio by a common factor.**

**Example 4:**

**A school has 300 students, of which 200 are girls and 100 are boys. **

**The ratio of girls to boys is 2:1, ****because there are twice as many girls as there are boys.**

**Equivalent Ratios**

**Equivalent ratios are ratios that have the same value but may look different. **

**We can create equivalent ratios by multiplying or dividing both sides of a ratio by the same number.**

**Example 1:**

**A rectangular room has a length of 12 feet and a width of 6 feet, which we can express as a ratio of 12:6. **

**We can simplify this ratio by dividing both sides by 6: ****12:6 = 2:1**

**This means that the length is twice as long as the width, ****but the overall size of the room has not changed.**

**Reduced Or lowest form of ratio**

**The reduced form of a ratio is the simplest possible form of the ratio. **

**To reduce a ratio, we divide both sides by their greatest common factor (GCF). **

**The GCF is the largest number that divides evenly into both sides of the ratio.**

**Example 1:**

**The ratio of red to blue marbles is 10:15. ****To reduce this ratio to its simplest form**

**we need to find the GCF of 10 and 15, which is 5. **

**Then we divide both sides of the ratio by 5:**

**10 ÷ 5 : 15 ÷ 5**

**2:3**

**So, reduced form of the ratio is 2:3.**

**Example 2:**

**Suppose a recipe calls for 3 cups of flour and 6 cups of sugar, ****which can be expressed as a ratio of 3:6. **

**To reduce this ratio, ****we need to find the GCF of 3 and 6, which is 3. ****Then we divide both sides of the ratio by 3:**

**3 ÷ 3 : 6 ÷ 3**

**1:2**

**So,the reduced form of the ratio is 1:2.**

**The ratio of boys to girls in a classroom is 8:12. ****To reduce this ratio, ****we need to find the GCF of 8 and 12, which is 4. **

**Then we divide both sides of the ratio by 4:**

**8 ÷ 4 : 12 ÷ 4**

**2:3**

**So,the reduced form of the ratio is 2:3.**

**Note:**

**Reducing ratios to their simplest form makes them easier to compare and work with. ****However, the actual quantities represented by the ratio do not change when we reduce the ratio. ****Only the way we express the relationship between the quantities changes.**

**Increased and Decreased Ratio**

**When the first quantity in a ratio is increased, it means that the ratio will have a larger numerator. **

**Conversely, when the first quantity is decreased, the ratio will have a smaller numerator. **

**This can also be expressed as a percentage increase or decrease.**

**Question 1: **

**The ratio of boys to girls in a class is 3:5. If the number of boys increases by 6 and the number of girls increases by 10, what is the new ratio of boys to girls?**

**Solution:**

**Start with the original ratio: 3:5**

**Increase the number of boys by 6: 3 + 6 = 9**

**Increase the number of girls by 10: 5 + 10 = 15**

**The new ratio is 9:15, which can be simplified to 3:5.**

**Requested by: Hamza**

**Alternative Method **

**Question 1.1: The ratio of boys to girls in a class is 3:5. If the number of boys increases by 15 and the number of girls increases by 25, what will be the new ratio of boys to girls?**

**Solution:**

**Let the number of boys be 3x and the number of girls be 5x.**

**Then, the current ratio of boys to girls is:**

**Boys : Girls = 3x : 5x**

**Simplifying, we get:**

**Boys : Girls = 3 : 5**

**If the number of boys increases by 15, then the new number of boys will be:**

**New number of boys = 3x + 15**

**If the number of girls increases by 25, then the new number of girls will be:**

**New number of girls = 5x + 25**

**The new ratio of boys to girls will be:**

**New Boys : New Girls = (3x + 15) : (5x + 25)**

**Simplifying, we get:**

**New Boys : New Girls = 3 : 5**

**Therefore, the new ratio of boys to girls remains the same, even though the actual number of boys and girls has increased.**

**Requested by: Hamza**

**Question 2:**

**In a mixture of milk and water, the ratio of milk to water is 2:1. ****If 5 liters of water is added to the mixture, what will be the new ratio of milk to water?**

**Solution:**

**Start with the original ratio: 2:1**

**Since only water is being added, we can assume that the amount of milk remains the same.**

**If the original amount of milk was 2x, then the original amount of water was x.**

**After 5 liters of water is added, the new amount of water is x+5.**

**The new ratio is 2x:(x+5).**

**We can solve for x by setting up the equation: 2x/x = 2/1. Solving for x, we get x = 5.**

**Therefore, the new ratio is 2(5):(5+5) = 10:10 or simplified to 1:1.**

**Requested by: Asadullah**

**Question 3: **

**The ratio of the number of employees in two companies is 5:7. ****If the first company hires 20 new employees and the second company hires 30 new employees, what is the new ratio of employees in the two companies?**

**Solution:**

**Start with the original ratio: 5:7**

**Increase the number of employees in the first company by 20: 5+20 = 25**

**Increase the number of employees in the second company by 30: 7+30 = 37**

**The new ratio is 25:37, which cannot be simplified any further.**

**Requested by: Asadullah**

**Question 4: **

**The ratio of the length to the width of a rectangle is 3:2.****If the length is increased by 6 and the width is increased by 4, what is the new ratio of the length to the width?**

**Solution:**

**Start with the original ratio: 3:2**

**Increase the length by 6: 3+6 = 9**

**Increase the width by 4: 2+4 = 6**

**Requested by: Muhammad Ali**

**Question 5:**

**The ratio of the number of red balls to blue balls in a bag is 2:3. ****If there are 20 more blue balls than red balls, how many balls are in the bag?**

**Solution:**

**Start with the original ratio: 2:3**

**Let the number of red balls be 2x.**

**Then, the number of blue balls is 3x.**

**Since there are 20 more blue balls than red balls, we can set up the equation: 3x – 2x = 20, which simplifies to x = 20.**

**Therefore, the number of red balls is 2x = 40 and the number of blue balls is 3x = 60. ****The total number of balls in the bag is 40 + 60 = 100.**

**Requested by: Ibrahim**

**Question 6: **

**The ratio of the perimeter of two similar triangles is 2:3. ****If the perimeter of the smaller triangle is 12 cm, what is the perimeter of the larger triangle?**

**Solution:**

**Start with the original ratio: 2:3**

**Let the perimeter of the smaller triangle be 2x.**

**Then, the perimeter of the larger triangle is 3x.**

**We can set up the equation: 2x = 12, which simplifies to x = 6.**

**Therefore, the perimeter of the larger triangle is 3x = 18 cm.**

**Requested by: Zaid**

**Question 7: **

**The ratio of the ages of two brothers is 3:5. If the difference in their ages is 10 years, what is the age of the younger brother?**

**Solution:**

**Start with the original ratio: 3:5**

**Let the age of the younger brother be 3x.**

**Then, the age of the older brother is 5x.**

**Since the difference in their ages is 10 years, we can set up the equation: 5x – 3x = 10, which simplifies to x = 5.**

**Therefore, the age of the younger brother is 3x = 15 years.**

**Requested by: Dua**

**Click the Link Below For Practice Problems**

**On increased ratios**

**Decreased In a Ratio**

**Question 1:**

**A store has a sale where the price of an item is reduced by 20%. ****If the original price of the item was $50, what is the new price?**

**Solution**

**If the price of an item is reduced by 20%, then the new price is 80% of the original price.**

**Therefore, the new price can be found by multiplying the original price by 0.8:**

**New price = $50 x 0.8 = $40.**

**Requested by: Aamir**

**Question 2:**

**The ratio of boys to girls in a class is 3:4. ****If there are 24 students in the class, how many of them are boys?**

**Solution**

**The ratio of boys to girls in the class is 3:4, **

**which means that for every 3 boys, there are 4 girls.**

**To find out how many boys there are in the class, we can use the following proportion:**

**3/7 = x/24**

**Cross-multiplying, we get:**

**3 x 24 = 7x**

**Simplifying, we get:**

**72 = 7x**

**Dividing both sides by 7, we get:**

**x = 10.29 (rounded to the nearest whole number)**

**Therefore, there are 10 boys in the class.**

**Requested by: Asma**

**Question 3:**

**A container is filled with a mixture of water and juice in the ratio 5:2. ****If there are 35 cups of the mixture in the container, how many cups of water are there?**

**Solution:**

**The ratio of water to the total mixture is 5/(5+2) = 5/7.**

**Therefore, there are 5/7 x 35 = 25 cups of water in the mixture.**

**Requested by:**

**Question 4:**

**The ratio of the length of a rectangle to its width is 4:3. **

**If the width of the rectangle is 6 cm, what is the length of the rectangle?**

**Solution:**

**The ratio of the length to the width is 4:3, which means that for every 4 units of length, there are 3 units of width.**

**If the width is 6 cm, then the length is:**

**4/3 x 6 = 8 cm.**

**Requested by: Anjali**

**Question 5:**

**A company’s profits decreased from $20 million to $16 million. ****What is the percentage decrease in profits?**

**Solution:**

**The decrease in profits is $20 million – $16 million = $4 million.**

**To find the percentage decrease, we can use the following formula:**

**Percentage decrease = (Decrease / Original value) x 100%**

**Substituting the values, we get:**

**Percentage decrease = ($4 million / $20 million) x 100% = 20%.**

**Therefore, the percentage decrease in profits is 20%.**

**Requested by: Asadullah**

**Question 6:**

**The ratio of the length of a rectangle to its width is 7:2. If the width of the rectangle is 4 cm, what is the length of the rectangle?**

**Solution:**

**The ratio of the length to the width is 7:2, which means that for every 7 units of length, there are 2 units of width.**

**If the width is 4 cm, then the length is:**

**7/2 x 4 = 14 cm.**

**Requested by: Omar**

**Question 7:**

**A restaurant had 100 customers on Monday and 80 customers on Tuesday. What is the percentage decrease in the number of customers?**

**Solution:**

**The decrease in the number of customers is 100 – 80 = 20.**

**To find the percentage decrease, we can use the following formula:**

**Percentage decrease = (Decrease / Original value) x 100%**

**Substituting the values, we get:**

**Percentage decrease = (20 / 100) x 100% = 20%.**

**Therefore, the percentage decrease in the number of customers is 20%.**

**Requested by: Asadullah**

**Click the Link Below For Practice Problems**

**On decreased ratios**

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