in marchn2017 joseph spent 60% of his income on house rent shopping and food in the ration 5:3:2 in june 2017 he spent 70% of his income on house rent shopping and food  in the ratio 6:5:3.the amount that he spent on shopping in june 2017 is on sixth less than what he spent in march 2017. find the percentage change in his income from march 2017 to june 2017?

Let's break down the problem step by step to find the percentage change in Joseph's income from March 2017 to June 2017.

### Given:
1. In March 2017:
   - Joseph spent 60% of his income on house rent, shopping, and food in the ratio 5:3:2.
2. In June 2017:
   - Joseph spent 70% of his income on house rent, shopping, and food in the ratio 6:5:3.
3. The amount spent on shopping in June 2017 is one-sixth less than the amount spent in March 2017.

### Step 1: Define Variables
Let:
- \( I_{\text{March}} \) be Joseph's income in March 2017.
- \( I_{\text{June}} \) be Joseph's income in June 2017.

### Step 2: Calculate Spending on Shopping in March 2017
In March 2017, the spending ratio for house rent, shopping, and food is 5:3:2.

The total expenditure in March is 60% of his income, so:
\[
\text{Total spending in March} = 0.6 \times I_{\text{March}}
\]
The amount spent on shopping in March is:
\[
\text{Shopping in March} = \frac{3}{10} \times 0.6 \times I_{\text{March}} = 0.18 \times I_{\text{March}}
\]

### Step 3: Calculate Spending on Shopping in June 2017
In June 2017, the spending ratio for house rent, shopping, and food is 6:5:3.

The total expenditure in June is 70% of his income, so:
\[
\text{Total spending in June} = 0.7 \times I_{\text{June}}
\]
The amount spent on shopping in June is:
\[
\text{Shopping in June} = \frac{5}{14} \times 0.7 \times I_{\text{June}} = 0.25 \times I_{\text{June}}
\]

### Step 4: Relationship Between March and June Spending on Shopping
It's given that the amount spent on shopping in June 2017 is one-sixth less than what was spent in March 2017:
\[
\text{Shopping in June} = \text{Shopping in March} - \frac{1}{6} \times \text{Shopping in March}
\]

Substitute the expressions for Shopping in March and Shopping in June:
\[
0.25 \times I_{\text{June}} = 0.18 \times I_{\text{March}} - \frac{1}{6} \times 0.18 \times I_{\text{March}}
\]
Simplify:
\[
0.25 \times I_{\text{June}} = 0.18 \times I_{\text{March}} \times \left(1 - \frac{1}{6}\right)
\]
\[
0.25 \times I_{\text{June}} = 0.18 \times I_{\text{March}} \times \frac{5}{6}
\]
\[
0.25 \times I_{\text{June}} = 0.15 \times I_{\text{March}}
\]

### Step 5: Calculate the Ratio of Incomes
Solve for the ratio \( \frac{I_{\text{June}}}{I_{\text{March}}} \):
\[
\frac{I_{\text{June}}}{I_{\text{March}}} = \frac{0.15}{0.25} = \frac{3}{5}
\]

So, \( I_{\text{June}} = \frac{3}{5} \times I_{\text{March}} \).

### Step 6: Find the Percentage Change in Income
The percentage change in income from March 2017 to June 2017 is:
\[
\text{Percentage Change} = \left(\frac{I_{\text{June}} - I_{\text{March}}}{I_{\text{March}}}\right) \times 100 = \left(\frac{\frac{3}{5}I_{\text{March}} - I_{\text{March}}}{I_{\text{March}}}\right) \times 100
\]
\[
\text{Percentage Change} = \left(\frac{-\frac{2}{5}I_{\text{March}}}{I_{\text{March}}}\right) \times 100 = -\frac{2}{5} \times 100 = -40\%
\]

### Final Answer
The percentage change in Joseph's income from March 2017 to June 2017 is a **40% decrease**.

Learn how to calculate the percentage change in Joseph's income from March 2017 to June 2017. Understand the mathematical concepts of percentage change, ratios, and income calculation through a detailed step-by-step solution. Suitable for students in Grades 7-9.

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