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The problem in the image involves determining the total mass of a bag of sugar. Here's the information extracted from the image:

1. The bag of sugar weighs less than 5.5 kg but more than 4.7 kg.
2. The bag was divided into two portions:
   - The first portion consists of several small packages, each weighing 500 g, with a total remainder of 400 g. This means the total weight of this portion is \( n \times 500 \, \text{g} + 400 \, \text{g} \).
   - The second portion also consists of several small packages, each weighing 600 g, with a total remainder of 100 g. This means the total weight of this portion is \( m \times 600 \, \text{g} + 100 \, \text{g} \).

We need to determine the total mass of the bag of sugar in kilograms.

### Let's solve the problem step by step:

1. Convert the remainders from grams to kilograms:
   - 400 g = 0.4 kg
   - 100 g = 0.1 kg

2. Let's express the total mass of the sugar as the sum of the two portions:
   \[
   \text{Total mass} = \left( n \times 500 \, \text{g} + 400 \, \text{g} \right) + \left( m \times 600 \, \text{g} + 100 \, \text{g} \right)
   \]
   Converting the weights to kilograms:
   \[
   \text{Total mass} = \left( n \times 0.5 \, \text{kg} + 0.4 \, \text{kg} \right) + \left( m \times 0.6 \, \text{kg} + 0.1 \, \text{kg} \right)
   \]
   Simplifying further:
   \[
   \text{Total mass} = 0.5n + 0.4 + 0.6m + 0.1 \, \text{kg}
   \]
   \[
   \text{Total mass} = 0.5n + 0.6m + 0.5 \, \text{kg}
   \]

3. According to the problem, the total mass lies between 4.7 kg and 5.5 kg:
   \[
   4.7 \leq 0.5n + 0.6m + 0.5 \leq 5.5
   \]

4. Subtract 0.5 from all parts of the inequality:
   \[
   4.2 \leq 0.5n + 0.6m \leq 5.0
   \]

5. Now, find integer values of \( n \) and \( m \) that satisfy the inequality:
   - \( 0.5n + 0.6m \) must be between 4.2 and 5.0.

6. To check possible values of \( n \) and \( m \):
   - For \( n = 4 \) and \( m = 5 \):
     \[
     0.5(4) + 0.6(5) = 2.0 + 3.0 = 5.0 \, \text{kg}
     \]
   - The total mass in this case is \( 5.0 + 0.5 = 5.5 \, \text{kg} \).
   - However, 5.5 kg does not satisfy the condition of being less than 5.5 kg.

   - For \( n = 4 \) and \( m = 4 \):
     \[
     0.5(4) + 0.6(4) = 2.0 + 2.4 = 4.4 \, \text{kg}
     \]
   - The total mass in this case is \( 4.4 + 0.5 = 4.9 \, \text{kg} \), which satisfies the condition.

Thus, the total mass of the sugar is **4.9 kg**.

Would you like further details or have any questions?

### Related Questions:
1. What would the total mass be if the remainders were different?
2. How would the solution change if the range was different, say between 4.5 kg and 5.3 kg?
3. Can this method be applied to other types of measurement units?
4. What if the small packages had different weights in both portions?
5. How does changing the number of small packages affect the total mass?

### Tip:
When dealing with inequalities, always check the boundary values to ensure they satisfy the given conditions.

The shopkeeper of a mini market bought a bag of sugar, and its mass is less than 5.5 kg but more than 4.7 kg. He divided it into two portions: one with small packages of 500g each and a remainder of 400g, and the other with small packages of 600g each and a remainder of 100g. What is the total mass of the bag of sugar in kg?

Learn how to calculate the total mass of a bag of sugar with detailed step-by-step algebraic solutions. The problem involves determining the mass by dividing it into two portions with specific small packages and remainders. Suitable for students in Grades 7-9.

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