Sharing in a Ratio
What is a Ratio?
A ratio is a way to compare two or more quantities. It shows the relative size of one quantity to another. For example, if a fruit salad recipe uses 2 cups of apples and 3 cups of grapes, the ratio of apples to grapes is 2:3. This means for every 2 parts of apples, there are 3 parts of grapes. Ratios can be written in different forms, such as with a colon (2:3), as a fraction ($\frac{2}{3}$), or with the word "to" (2 to 3).
Ratios are used to describe how a whole is divided into parts. When we share in a ratio, we are taking a total amount and splitting it up so that the portions reflect this comparative relationship. The key is that the shares are proportional; they maintain the same relationship to each other as the numbers in the ratio.
The Step-by-Step Method for Sharing in a Ratio
Sharing a quantity in a ratio can be broken down into a simple, reliable process. Let's use the example of sharing $50 between two people, Alex and Ben, in the ratio 3:2.
Step 1: Add the parts of the ratio.
Add the numbers in the ratio to find the total number of parts. For the ratio 3:2, the total parts are $3 + 2 = 5$.
Step 2: Find the value of one part.
Divide the total quantity by the total number of parts. $\frac{\$50}{5} = \$10$. So, one part is worth $10.
Step 3: Multiply to find each share.
Multiply the value of one part by each number in the ratio.
- Alex's share: $3 \times \$10 = \$30$
- Ben's share: $2 \times \$10 = \$20$
You can always check your work by adding the shares together to make sure they equal the original total: $\$30 + \$20 = \$50$.
| Step | Action | Example: Share $50 in ratio 3:2 |
|---|---|---|
| 1 | Add the ratio parts | $3 + 2 = 5$ total parts |
| 2 | Find value of one part | $\frac{\$50}{5} = \$10$ per part |
| 3 | Multiply for each share | Alex: $3 \times \$10 = \$30$ Ben: $2 \times \$10 = \$20$ |
| 4 | Check your work | $\$30 + \$20 = \$50$ ✓ |
Working with More Than Two Parts
The method is exactly the same when sharing between three or more parts. Suppose a prize of $120 is to be shared between three winners, Anna, Bella, and Chloe, in the ratio 3:2:1.
Step 1: Total parts = $3 + 2 + 1 = 6$
Step 2: Value of one part = $\frac{\$120}{6} = \$20$
Step 3: Find each share:
- Anna: $3 \times \$20 = \$60$
- Bella: $2 \times \$20 = \$40$
- Chloe: $1 \times \$20 = \$20$
Step 4: Check: $\$60 + \$40 + \$20 = \$120$
Simplifying Ratios Before Sharing
Sometimes, the ratio you are given can be simplified, just like a fraction. This makes the numbers smaller and easier to work with. To simplify a ratio, divide all the parts by the same number. For example, the ratio 6:4:2 can be simplified by dividing each part by 2, giving the simpler ratio 3:2:1. The proportional relationship is exactly the same. It is always a good idea to simplify the ratio first if possible.
Real-World Applications of Ratio Sharing
Sharing in a ratio is not just a math class exercise; it is a skill used constantly in daily life and various professions.
In Everyday Life:
- Splitting a Bill: If three friends eat out and agree to split the $75 bill in the ratio of what they ordered (e.g., 2:2:1), they can use this method to find out how much each owes fairly.
- Scaling Recipes: If a cookie recipe for 10 people uses flour and sugar in a 3:1 ratio, and you need to make it for 25 people, you can scale up the ingredients while maintaining the correct ratio.
- Mixing Concrete or Paint: DIY projects often require mixing components in specific ratios, like cement, sand, and gravel in a 1:2:3 ratio to make concrete.
In Business and Finance:
- Profit Sharing: If three partners invest different amounts of money into a business, they will often share the profits in the same ratio as their investments. An investor who put in twice as much money as another gets twice the share of the profits.
- Allocating Resources: A school might allocate its budget for sports equipment to different sports teams based on the ratio of team members.
In Science:
- Chemical Compounds: Many chemical compounds are formed when elements combine in fixed ratios. For example, the water molecule H2O has a ratio of 2:1 for hydrogen to oxygen atoms.
- Drug Dosages: Medicine is often prescribed based on a patient's weight, establishing a ratio of milligrams of drug per kilogram of body weight.
Solving Different Types of Ratio Problems
Sometimes, you are not given the total amount to be shared. You might be given the value of one share or the difference between two shares. The problem-solving strategy adjusts slightly for these cases.
Example 1: Given the value of one part.
A sum of money is shared between Tom and Jerry in the ratio 5:4. If Jerry gets $28, how much does Tom get, and what was the total amount?
Jerry's share corresponds to 4 parts. So, 4 parts = $28.
Value of one part = $\frac{\$28}{4} = \$7$.
Tom's share (5 parts) = $5 \times \$7 = \$35$.
Total amount = $\$35 + \$28 = \$63$.
Example 2: Given the difference between shares.
Money is shared between two sisters in the ratio 7:5. If the older sister receives $30 more than the younger sister, what was the total shared?
The difference in the ratio parts is $7 - 5 = 2$ parts.
This difference is equal to $30. So, 2 parts = $30.
Value of one part = $\frac{\$30}{2} = \$15$.
Total parts = $7 + 5 = 12$.
Total amount = $12 \times \$15 = \$180$.
Common Mistakes and Important Questions
Q: What is the difference between a ratio and a fraction?
A ratio compares parts to parts, while a fraction compares a part to the whole. In the ratio 3:2, the 3 and 2 are the parts. The fraction of the whole that the first part represents is $\frac{3}{3+2} = \frac{3}{5}$.
Q: What do I do if the ratio has decimals?
The best approach is to multiply every part of the ratio by the same number (like 10 or 100) to turn them into whole numbers first. For example, to share in the ratio 1.5 : 2.5, multiply both parts by 2 to get the equivalent ratio 3 : 5. Then proceed with the simplified whole-number ratio.
Q: What is the most common error when sharing in a ratio?
The most frequent mistake is forgetting to add the parts of the ratio to find the total number of parts. A student might see the ratio 2:3 and incorrectly divide the total by 2 or 3, instead of by 5. Always remember: Sum the parts, then divide the total.
Sharing in a ratio is a fundamental mathematical tool for fair and proportional division. The process is logical and consistent: find the total number of parts, determine the value of a single part, and then distribute the quantity accordingly. This skill transcends the classroom, proving invaluable in cooking, finance, science, and countless everyday situations. By mastering the step-by-step method and understanding how to apply it to different scenarios, you empower yourself to solve problems involving proportional distribution efficiently and accurately. Remember to always simplify ratios where possible and, most importantly, check that your final shares add up to the original total.
Footnote
[1] Proportional Relationship: A relationship between two quantities where the ratio of one quantity to the other is constant. This means that if you double one quantity, the other also doubles, maintaining the same ratio. This is the core idea behind sharing in a ratio.