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 denote Joseph's income in March 2017 by \( M \) and in June 2017 by \( J \).

### Step 1: Determine the amounts spent in March 2017
- Joseph spent 60% of his income on house rent, shopping, and food in the ratio 5:3:2.
- The total proportion is \( 5 + 3 + 2 = 10 \).

**Amount spent on shopping in March 2017:**
\[
\text{Shopping in March} = 60\% \times M \times \frac{3}{10} = 0.6M \times 0.3 = 0.18M
\]

### Step 2: Determine the amounts spent in June 2017
- Joseph spent 70% of his income on house rent, shopping, and food in the ratio 6:5:3.
- The total proportion is \( 6 + 5 + 3 = 14 \).

**Amount spent on shopping in June 2017:**
\[
\text{Shopping in June} = 70\% \times J \times \frac{5}{14} = 0.7J \times \frac{5}{14} = \frac{3.5J}{14} = 0.25J
\]

### Step 3: Relate the shopping expenses between March and June
We are told that the amount spent on shopping in June 2017 is one-sixth less than what he spent in March 2017.

This means:
\[
0.25J = 0.18M \times \left(1 - \frac{1}{6}\right) = 0.18M \times \frac{5}{6} = \frac{0.9M}{6} = 0.15M
\]

### Step 4: Solve for the ratio \( \frac{J}{M} \)
Now, equate the expressions for shopping:
\[
0.25J = 0.15M
\]
\[
\frac{J}{M} = \frac{0.15}{0.25} = \frac{3}{5}
\]

### Step 5: Calculate the percentage change in income
The percentage change in Joseph's income from March 2017 to June 2017 is:
\[
\text{Percentage change} = \left(\frac{J - M}{M}\right) \times 100 = \left(\frac{\frac{3}{5}M - M}{M}\right) \times 100 = \left(\frac{-2M}{5M}\right) \times 100 = -40\%
\]

This means Joseph's income **decreased by 40%** from March 2017 to June 2017.

Would you like further details on this solution, or do you have any questions?

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Here are 5 questions related to this problem:

1. What would be the percentage change in Joseph's income if the shopping expenses in June were one-fourth less than in March?
2. How much would Joseph have spent on house rent in June if his income stayed the same?
3. What if Joseph's income had increased by 20% from March to June, how would the shopping expense relate then?
4. If Joseph's total expenses remained constant between March and June, how would his income have changed?
5. How would the solution change if the ratios in March were instead 4:4:2?

**Tip:** When working with percentage changes, make sure to clearly identify the base value (initial value) to avoid confusion.

Explore how to calculate the percentage change in Joseph's income from March to June 2017 based on his expenses on house rent, shopping, and food. Understand the impact of spending ratios and percentage differences in this detailed math problem.

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