The number of units manufactured by a company A was 12500 units in 2019 and 10625 units in 2020. While in company B, the production fell from 34000 units in 2019 to 30600 units in 2020. If X and Y are the percentage decrease in the number of units manufactured by company A and B respectively from 2019 to 2020, then what will be the ratio of X and Y?
3 : 2
Understanding Percentage Decrease in Units Manufactured
This problem asks us to calculate the percentage decrease in the number of units manufactured for two different companies, A and B, from the year 2019 to 2020. After finding these percentage decreases, which are represented by X and Y for company A and B respectively, we need to determine the ratio of X to Y.
Calculating Percentage Decrease for Company A
First, let's focus on Company A. We are given the number of units manufactured in 2019 and 2020.
- Units manufactured by Company A in 2019 = 12500
- Units manufactured by Company A in 2020 = 10625
To find the decrease in the number of units, we subtract the 2020 production from the 2019 production.
Decrease in units for Company A = Units in 2019 - Units in 2020
Decrease for Company A = $12500 - 10625 = 1875$ units
The percentage decrease is calculated as the decrease divided by the original number of units (in 2019), multiplied by 100. This percentage decrease is given as X.
Percentage Decrease (X) = $\left( \frac{\text{Decrease}}{\text{Original Units}} \right) \times 100$
X = $\left( \frac{1875}{12500} \right) \times 100$
Now, let's simplify the calculation for X:
X = $\frac{1875}{125}$
X = $15$
So, the percentage decrease in the number of units manufactured by company A is 15%. Thus, X = 15.
Calculating Percentage Decrease for Company B
Next, let's calculate the percentage decrease for Company B.
- Units manufactured by Company B in 2019 = 34000
- Units manufactured by Company B in 2020 = 30600
Calculate the decrease in units for Company B:
Decrease in units for Company B = Units in 2019 - Units in 2020
Decrease for Company B = $34000 - 30600 = 3400$ units
The percentage decrease for Company B is given as Y.
Percentage Decrease (Y) = $\left( \frac{\text{Decrease}}{\text{Original Units}} \right) \times 100$
Y = $\left( \frac{3400}{34000} \right) \times 100$
Now, let's simplify the calculation for Y:
Y = $\left( \frac{3400}{34000} \right) \times 100 = \left( \frac{34}{340} \right) \times 100 = \left( \frac{1}{10} \right) \times 100$
Y = $10$
So, the percentage decrease in the number of units manufactured by company B is 10%. Thus, Y = 10.
Finding the Ratio of X and Y
We are asked to find the ratio of X and Y, which is X : Y.
Ratio X : Y = $15 : 10$
To simplify the ratio, we find the greatest common divisor (GCD) of 15 and 10, which is 5. We divide both numbers by 5.
Ratio X : Y = $\frac{15}{5} : \frac{10}{5}$
Ratio X : Y = $3 : 2$
The ratio of X and Y is 3 : 2.
Summary of Calculations
| Company | Units in 2019 | Units in 2020 | Decrease | Percentage Decrease |
|---|---|---|---|---|
| A | 12500 | 10625 | 1875 | $X = \left( \frac{1875}{12500} \right) \times 100 = 15\%$ |
| B | 34000 | 30600 | 3400 | $Y = \left( \frac{3400}{34000} \right) \times 100 = 10\%$ |
Ratio of X and Y = $15 : 10 = 3 : 2$.
Revision Table: Percentage Change Concepts
| Concept | Formula | Notes |
|---|---|---|
| Percentage Increase | $\left( \frac{\text{New Value} - \text{Original Value}}{\text{Original Value}} \right) \times 100$ | Used when the new value is greater than the original. |
| Percentage Decrease | $\left( \frac{\text{Original Value} - \text{New Value}}{\text{Original Value}} \right) \times 100$ | Used when the new value is less than the original. |
| Change (Increase/Decrease) | $|\text{New Value} - \text{Original Value}|$ | Absolute difference between values. |
Additional Information on Ratios and Percentages
Understanding ratios and percentages is fundamental in solving quantitative problems like this. A ratio compares two quantities, while a percentage expresses a quantity as a fraction of 100.
- Ratio: A ratio $a:b$ can be written as $\frac{a}{b}$. It shows the relative sizes of two or more values. Ratios can be simplified by dividing all parts by their greatest common divisor, as we did with 15:10.
- Percentage: Percentage is calculated as $\frac{\text{Part}}{\text{Whole}} \times 100$. In percentage change calculations, the 'Whole' is always the original value before the change occurred. This is crucial to remember.
In this problem, X and Y represent percentage decreases, calculated relative to the production in 2019 for each company. We found X = 15% and Y = 10%. The ratio of these percentages is simply the ratio of the numbers 15 and 10, which simplifies to 3:2.
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