2.4 CHAPTER 2. EQUATIONS AND INEQUA $$ { }_{4} $$ Exercise 2-4 1. Two jets are flying towards each other from airports that are 1200 km apart. One jet is flying at $$ 250 \mathrm{~km} / \mathrm{h} $$ and the other jet at $$ 350 \mathrm{~km} / \mathrm{h} $$. If they took off at the same time, how long will it take for the jets to pass each other? 2. Kadesh bought 20 shirts at a total cost of $$ R 980 $$. If the large shirts cost R 50 and the small shirts cost R 40 . How many of each size did he buy? 3. The diagonal of a rectangle is 25 cm more than its width. The length of the rectangle is 17 cm more than its width. What are the dimensions of the rectangle? 4. The sum of 27 and 12 is equal to 73 more than an unknown number. Find the unknown number. 5. The two smaller angles in a right-angled triangle are in the ratio of $$ 1: 2 $$. What are the sizes of the two angles? 6. The length of a rectangle is twice the breadth. If the area is $$ 128 \mathrm{~cm}^{2} $$, determine the length and the breadth. 7. If 4 times a number is increased by 6 , the result is 15 less than the square of the number. Find the number. 8. The length of a rectangle is 2 cm more than the width of the rectangle. The perimeter of the rectangle is 20 cm . Find the length and the width of the rectangle. 9. Stephen has $$ 1 \ell $$ of a mixture containing $$ 69 \% $$ salt. How much water must Stephen add to make the mixture $$ 50 \% $$ salt? Write your answer as a fraction of a litre. 10. The sum of two consecutive odd numbers is 20 and $$ t $$ 2. Find the two numbers. 11. The denominator of a fraction is

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Let's solve each of the problems step by step.

### Problem 1
**Two jets are flying towards each other from airports that are 1200 km apart. One jet is flying at $$ 250 \mathrm{~km/h} $$ and the other jet at $$ 350 \mathrm{~km/h} $$. If they took off at the same time, how long will it take for the jets to pass each other?**

**Known Conditions:**
- Distance between jets: $$ D = 1200 $$ km
- Speed of jet 1: $$ S_1 = 250 $$ km/h
- Speed of jet 2: $$ S_2 = 350 $$ km/h

**Step 1: Calculate the combined speed of the jets.**
$$
S_{total} = S_1 + S_2 = 250 + 350 = 600 \text{ km/h}
$$

**Step 2: Use the formula for time, $$ t = \frac{D}{S} $$.**
$$
t = \frac{D}{S_{total}} = \frac{1200}{600} = 2 \text{ hours}
$$

### Problem 2
**Kadesh bought 20 shirts at a total cost of $$ R 980 $$. If the large shirts cost R 50 and the small shirts cost R 40. How many of each size did he buy?**

**Known Conditions:**
- Total shirts: $$ x + y = 20 $$
- Total cost: $$ 50x + 40y = 980 $$

**Step 1: Solve the first equation for $$ y $$.**
$$
y = 20 - x
$$

**Step 2: Substitute $$ y $$ in the second equation.**
$$
50x + 40(20 - x) = 980
$$
$$
50x + 800 - 40x = 980
$$
$$
10x = 180 \implies x = 18
$$
$$
y = 20 - 18 = 2
$$

**Kadesh bought 18 large shirts and 2 small shirts.**

### Problem 3
**The diagonal of a rectangle is 25 cm more than its width. The length of the rectangle is 17 cm more than its width. What are the dimensions of the rectangle?**

**Known Conditions:**
- Let width = $$ w $$
- Length = $$ w + 17 $$
- Diagonal = $$ w + 25 $$

**Step 1: Use the Pythagorean theorem.**
$$
(w + 17)^2 + w^2 = (w + 25)^2
$$

**Step 2: Expand and simplify.**
$$
w^2 + 34w + 289 + w^2 = w^2 + 50w + 625
$$
$$
2w^2 + 34w + 289 = w^2 + 50w + 625
$$
$$
w^2 - 16w - 336 = 0
$$

**Step 3: Solve the quadratic equation.**
Using the quadratic formula $$ w = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$:
$$
w = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot (-336)}}{2 \cdot 1}
$$
$$
w = \frac{16 \pm \sqrt{256 + 1344}}{2}
$$
$$
w = \frac{16 \pm \sqrt{1600}}{2} = \frac{16 \pm 40}{2}
$$
$$
w = 28 \text{ or } w = -12 \text{ (not valid)}
$$

**Step 4: Find length and diagonal.**
$$
l = w + 17 = 28 + 17 = 45
$$
$$
d = w + 25 = 28 + 25 = 53
$$

**Dimensions of the rectangle: Width = 28 cm, Length = 45 cm.**

### Problem 4
**The sum of 27 and 12 is equal to 73 more than an unknown number. Find the unknown number.**

**Known Conditions:**
$$
27 + 12 = x + 73
$$

**Step 1: Simplify the equation.**
$$
39 = x + 73
$$
$$
x = 39 - 73 = -34
$$

**The unknown number is -34.**

### Problem 5
**The two smaller angles in a right-angled triangle are in the ratio of $$ 1: 2 $$. What are the sizes of the two angles?**

**Known Conditions:**
- Let the angles be $$ x $$ and $$ 2x $$.
- Since it's a right triangle, $$ x + 2x + 90 = 180 $$.

**Step 1: Solve for $$ x $$.**
$$
3x + 90 = 180
$$
$$
3x = 90 \implies x = 30
$$

**Step 2: Find the angles.**
$$
\text{Angles: } 30^\circ \text{ and } 60^\circ
$$

### Problem 6
**The length of a rectangle is twice the breadth. If the area is $$ 128 \mathrm{~cm}^2 $$, determine the length and the breadth.**

**Known Conditions:**
- Let breadth = $$ b $$
- Length = $$ 2b $$
- Area = $$ 2b \cdot b = 128 $$

**Step 1: Set up the equation.**
$$
2b^2 = 128 \implies b^2 = 64 \implies b = 8
$$

**Step 2: Find length.**
$$
l = 2b = 16
$$

**Dimensions: Length = 16 cm, Breadth = 8 cm.**

### Problem 7
**If 4 times a number is increased by 6, the result is 15 less than the square of the number. Find the number.**

**Known Conditions:**
- Let the number be $$ x $$.
- Equation: $$ 4x + 6 = x^2 - 15 $$

**Step 1: Rearrange the equation.**
$$
x^2 - 4x - 21 = 0
$$

**Step 2: Factor the quadratic.**
$$
(x - 7)(x + 3) = 0
$$
$$
x = 7 \text{ or } x = -3
$$

**The number can be 7 or -3.**

### Problem 8
**The length of a rectangle is 2 cm more than the width of the rectangle. The perimeter of the rectangle is 20 cm. Find the length and the width of the rectangle.
**Length = 6 cm, Width = 4 cm.**

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