1. Three consecutive natural numbers add up to 24 . Find the numbers. 2. A mother is three times as old as her daughter. In 12 years' time she will be twice as old as her daughter will be then. How old is the mother now? 3. The perimeter of a rectangular flower bed is 26 m . If the length exceeds the breadth by 3 m , find its dimenin

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**1. Three consecutive natural numbers add up to 24**

Let the numbers be $$ n $$, $$ n+1 $$, and $$ n+2 $$.

The equation becomes:
$$
n + (n+1) + (n+2) = 24
$$
Simplify:
$$
3n + 3 = 24
$$
Subtract 3 from both sides:
$$
3n = 21
$$
Divide both sides by 3:
$$
n = 7
$$
Thus, the three numbers are:
$$
7,\quad 8,\quad 9
$$

---

**2. A mother is three times as old as her daughter. In 12 years' time she will be twice as old as her daughter will be then. How old is the mother now?**

Let the daughter's age be $$ d $$ and the mother's age be $$ m $$.

Given:
$$
m = 3d
$$
In 12 years, the ages will be $$ m+12 $$ and $$ d+12 $$, and it is given that:
$$
m+12 = 2(d+12)
$$
Substitute $$ m = 3d $$ into the equation:
$$
3d + 12 = 2d + 24
$$
Subtract $$ 2d $$ from both sides:
$$
d + 12 = 24
$$
Subtract 12 from both sides:
$$
d = 12
$$
Now, find $$ m $$:
$$
m = 3 \times 12 = 36
$$

Thus, the mother is:
$$
36 \text{ years old}
$$

---

**3. The perimeter of a rectangular flower bed is 26 m. If the length exceeds the breadth by 3 m, find its dimensions.**

Let the breadth be $$ b $$ and the length be $$ b+3 $$.

The perimeter $$ P $$ of a rectangle is given by:
$$
P = 2(\text{length} + \text{breadth})
$$
Substitute the given values:
$$
2(b + (b+3)) = 26
$$
Simplify inside the parentheses:
$$
2(2b+3) = 26
$$
Divide both sides by 2:
$$
2b + 3 = 13
$$
Subtract 3 from both sides:
$$
2b = 10
$$
Divide by 2:
$$
b = 5
$$
Now, the length is:
$$
b+3 = 5+3 = 8
$$

Thus, the dimensions of the rectangular flower bed are:
$$
\text{Breadth} = 5\, \text{m} \quad \text{and} \quad \text{Length} = 8\, \text{m}
$$
1. The three consecutive natural numbers are 7, 8, and 9. 2. The mother is 36 years old now. 3. The rectangular flower bed has a breadth of 5 meters and a length of 8 meters.

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