A family has 480kg of rice for X number of weeks. If they need to use the same amount for 4 more weeks, they need to cut down their weekly assumption of rice by 4kg. What is the value of X?

Let's solve this problem step by step.

1. **Define the variables:**
   - Let \( X \) be the original number of weeks.
   - Let \( R \) be the original weekly consumption of rice in kilograms.

2. **Write down the equations:**
   - The total amount of rice is \( 480 \) kg.
   - The total consumption over \( X \) weeks is \( X \times R = 480 \).

   When the consumption is reduced by 4 kg per week, the family can extend the rice supply for 4 more weeks:
   - The new number of weeks is \( X + 4 \).
   - The new weekly consumption is \( R - 4 \) kg.

   Therefore, the new equation is:
   \[
   (X + 4) \times (R - 4) = 480
   \]

3. **Substitute \( R \) from the first equation:**
   From the first equation, \( R = \frac{480}{X} \).

   Substitute \( R \) in the second equation:
   \[
   (X + 4) \times \left(\frac{480}{X} - 4\right) = 480
   \]

4. **Simplify and solve the equation:**
   Expand and simplify:
   \[
   (X + 4) \times \left(\frac{480 - 4X}{X}\right) = 480
   \]
   Multiply through by \( X \) to eliminate the fraction:
   \[
   (X + 4)(480 - 4X) = 480X
   \]

   Expand the left side:
   \[
   480X - 4X^2 + 1920 - 16X = 480X
   \]

   Simplify by canceling out \( 480X \) from both sides:
   \[
   -4X^2 - 16X + 1920 = 0
   \]
   Divide the entire equation by \(-4\) to simplify:
   \[
   X^2 + 4X - 480 = 0
   \]

5. **Solve the quadratic equation:**
   Use the quadratic formula:
   \[
   X = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   \]
   Here, \( a = 1 \), \( b = 4 \), and \( c = -480 \):
   \[
   X = \frac{-4 \pm \sqrt{4^2 + 1920}}{2}
   \]
   \[
   X = \frac{-4 \pm \sqrt{16 + 1920}}{2}
   \]
   \[
   X = \frac{-4 \pm \sqrt{1936}}{2}
   \]
   \[
   X = \frac{-4 \pm 44}{2}
   \]
   Therefore, the two possible solutions for \( X \) are:
   \[
   X = \frac{-4 + 44}{2} = 20
   \]
   \[
   X = \frac{-4 - 44}{2} = -24 \quad (\text{not valid, as \( X \) must be positive})
   \]

Thus, the value of \( X \) is \( 20 \) weeks.

Would you like further details or have any other questions?

Here are 5 related questions you might find interesting:
1. How much rice would the family consume weekly if they used it over \( 24 \) weeks instead?
2. What if the family had 600 kg of rice and wanted to extend its use by 6 weeks?
3. How would the weekly consumption change if the rice was for 30 weeks instead of 20?
4. What is the new weekly consumption if the family reduces the consumption by 3 kg instead of 4 kg to extend the weeks by 5?
5. How would the problem change if the family needed to extend the consumption by a different number of weeks?

**Tip:** When solving quadratic equations, always check if the discriminant \( b^2 - 4ac \) is positive to ensure real solutions exist.

Discover how to solve a quadratic equation to find the number of weeks a family can sustain their rice consumption. This problem involves key algebraic concepts, including linear and quadratic equations, suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation