Q: What is the total number of numbers less than \$1000\$ and divisible by \$5\$, which can be formed with digits \$0\$ to \$9\$, such that no digit gets repeated?

Condition \$1 :\$ Number \$< 1000\$.Condition \$2\$ : Number divisible by \$5\$Suppose it’s a \$1\$ digit number. Then only '5' falls under the above condition (1 way)Suppose it’s a \$2\$ digit number:\$ __ \$ \$__\$If second digit is \$0\$, no of ways to fill 1st digit \$\Rightarrow\$ \$1\$ to \$9\$. (\$9\$ ways)If second digit is \$5\$, no of ways to fill 1st digit \$\Rightarrow\$ \$1,2,3,4,6,7,8,9\$. (\$8\$ ways)Total ways : \$9 + 8 = 17\$Suppose it’s a \$3\$ digit number: \$__\$ \$ __\$ \$ __\$If last digit is \$0\$; \${1}^{st}\$ digit = \$9\$ ways, \${2}^{nd}\$ digit =\$8\$ ways, Total \$=8\times 9=72\$ waysIf last digit is \$5\$; \${1}^{st}\$ digit = \$8\$ ways, \${2}^{nd}\$ digit = \$8\$ ways, Total \$=8\times 8=64\$ waysTotal ways : \$72 + 64 = 136\$Total no of numbers :\$ 1 + 17 + 136 = 154 \$ ways.

Q: In a fraction, twice the numerator is \$2\$ more than the denominator.If \$3\$ is added to the numerator and to the denominator, the new fraction is \$\dfrac{2}{3}\$.Find the original fraction.

Let the numerator be \$x\$Denominator \$= 2x − 2\$Therefore the fraction \$= \dfrac{x}{2x − 2}\$Given: when \$3\$ is added to both the numerator and denominator, the fraction becomes \$\dfrac{2}{3}\$.Hence \$\dfrac{\left(x + 3\right)}{\left(2x − 2 + 3\right)} = \dfrac{2}{3}\$ \$\dfrac{\left(x + 3\right)}{\left(2x + 1\right)}= \dfrac{2}{3}\$\$\Rightarrow 3x + 9 = 4x + 2\$ \$\Rightarrow x = 7\$\$\therefore\$ the fraction is \$\dfrac{7}{12}\$.

Q: The value of question mark is :

Working in rows, the number in the centre equals the sum of the left and right hand numbers, divided by 3.In the first row, middle value is \$=\cfrac{5+7}{3}=\cfrac{12}{3}=4\$In the second row, middle value is \$=\cfrac{12+6}{3}=\cfrac{18}{3}=6\$Hence, required value = \$\cfrac{7+2}{3}=\cfrac{9}{3}=3\$

Q: Study the following pattern:\$ 1 = \dfrac {1 \times 2}{2}\$\$ 1+2 = \dfrac {2 \times 3}{2}\$\$ 1+2+3 = \dfrac {3 \times 4}{2}\$\$ 1+2+3+4 = \dfrac {4 \times 5}{2}\$By observing the above pattern, find \$ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 \$

Clearly the pattern being followed is :\$1+2+3+...+n=\dfrac{n(n+1)}{2}\$Now,\$ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 \$here, \$n=10\$So, \$1+2+3+4+5+6+7+8+9+10=\dfrac{10\times11}{2}=55\$Hence, the required value is \$55\$ .

Q: If abc-cba = 792, where abc is a 3-digit number, if a+b+c = 18 and a = 9c, what is the number?

We have, \$abc-cba=792\$\$abc\$ is a \$3-digit\$ number.Hence, \$abc=100a+10b+c\$Similarly, \$cba=100c+10b+a\$So, \$abc-bca=100a+10b+c-100c-10b-a\$\$=99a-99c\$\$=99(a-c)\$So, \$99(a-c)=792\$\$\Rightarrow a-c=8\$Also, \$a=9c\quad [given]\$\$\therefore\ 9c-c=8\$\$\Rightarrow 8c=8\$Hence, \$c=1\$So, \$a=9(1)\$Hence, \$a=9\$Also, we are given, \$a+b+c=18\$So, \$9+b+1=18\$\$\therefore \ b=8\$Hence, the number is \$981\$.

Q: 2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

Let number of rows = xEach row contain plants = xTotal plants = 2025Therefore,\$x^{2} = 2025\$\$x=45\$.

Question 1

What is the total number of numbers less than $1000$ and divisible by $5$, which can be formed with digits $0$ to $9$, such that no digit gets repeated?

Accepted Answer

Condition $1 :$ Number $< 1000$.Condition $2$ : Number divisible by $5$Suppose it&#8217;s a $1$ digit number. Then only '5' falls under the above condition (1 way)Suppose it&#8217;s a $2$ digit number:$ __ $ $__$If second digit is $0$, no of ways to fill 1st digit $\Rightarrow$ $1$ to $9$. ($9$ ways)If second digit is $5$, no of ways to fill 1st digit $\Rightarrow$ $1,2,3,4,6,7,8,9$. ($8$ ways)Total ways : $9 + 8 = 17$Suppose it&#8217;s a $3$ digit number: $__$ $ __$ $ __$If last digit is $0$; ${1}^{st}$ digit = $9$ ways, ${2}^{nd}$ digit =$8$ ways, Total $=8\times 9=72$ waysIf last digit is $5$; ${1}^{st}$ digit = $8$ ways, ${2}^{nd}$ digit = $8$ ways, Total $=8\times 8=64$ waysTotal ways : $72 + 64 = 136$Total no of numbers :$ 1 + 17 + 136 = 154 $ ways.

Question 2

In a fraction, twice the numerator is $2$ more than the denominator.If $3$ is added to the numerator and to the denominator, the new fraction is $\dfrac{2}{3}$.Find the original fraction.

Accepted Answer

Let the numerator be $x$Denominator $= 2x &#8722; 2$Therefore the fraction $= \dfrac{x}{2x &#8722; 2}$Given: when $3$ is added to both the numerator and denominator, the fraction becomes $\dfrac{2}{3}$.Hence $\dfrac{\left(x + 3\right)}{\left(2x &#8722; 2 + 3\right)} = \dfrac{2}{3}$ $\dfrac{\left(x + 3\right)}{\left(2x + 1\right)}= \dfrac{2}{3}$$\Rightarrow 3x + 9 = 4x + 2$ $\Rightarrow x = 7$$\therefore$ the fraction is $\dfrac{7}{12}$.

Question 3

Study the following pattern: $1+3   =2 \times 2$$1+ 3+5 = 3 \times 3$$ 1+ 3+7 = 4 \times4$$1+ 3 + 5 + 7 + 9 = 5 \times 5 $By observing the above pattern, find (i) $1+ 3 + 5 + 7+ 9 +11$(ii) $ 1+3+5+7+9+11+13+15 $(iii) $21+ 23 + 25 + + 51$

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Question 4

Write the first four products of two numbers with difference 4 starting from In the following order: 1, 2, 3, 4, 5, 6, ..... Identify the pattern in the products and write the next three products.

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Question 5

The value of question mark is :

Accepted Answer

Working in rows, the number in the centre equals the sum of the left and right hand numbers, divided by 3.In the first row, middle value is $=\cfrac{5+7}{3}=\cfrac{12}{3}=4$In the second row, middle value is $=\cfrac{12+6}{3}=\cfrac{18}{3}=6$Hence, required value = $\cfrac{7+2}{3}=\cfrac{9}{3}=3$

Question 6

Observe the pattern in the following and fill in the blanks:
$9 \times 9 + 7 = 88$
$98 \times 9 + 6 = 888$
$987 \times 9 + 5 = 8888$
$9876 \times 9 + 4 = . . . . . .$
$98765 \times 9 + 3 = . . . .$
$987654 \times 9 + 2 = . . . .$
$9876543 \times 9 + 1 = . . . . . .$

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Question 7

Study the following pattern:$ 1 = \dfrac {1 \times 2}{2}$$ 1+2 = \dfrac {2 \times 3}{2}$$ 1+2+3 = \dfrac {3 \times 4}{2}$$ 1+2+3+4 = \dfrac {4 \times 5}{2}$By observing the above pattern, find $ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 $

Accepted Answer

Clearly the pattern being followed is :$1+2+3+...+n=\dfrac{n(n+1)}{2}$Now,$ 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 $here, $n=10$So, $1+2+3+4+5+6+7+8+9+10=\dfrac{10\times11}{2}=55$Hence, the required value is $55$ .

Question 8

If abc-cba = 792, where abc is a 3-digit number, if a+b+c = 18 and a = 9c, what is the number?

Accepted Answer

We have, $abc-cba=792$$abc$ is a $3-digit$ number.Hence, $abc=100a+10b+c$Similarly, $cba=100c+10b+a$So, $abc-bca=100a+10b+c-100c-10b-a$$=99a-99c$$=99(a-c)$So, $99(a-c)=792$$\Rightarrow a-c=8$Also, $a=9c\quad [given]$$\therefore\ 9c-c=8$$\Rightarrow 8c=8$Hence, $c=1$So, $a=9(1)$Hence, $a=9$Also, we are given, $a+b+c=18$So, $9+b+1=18$$\therefore \ b=8$Hence, the number is $981$.

Question 9

Observe the following pattern and extend it to three more steps:
$6 \times 2 - 5 = 7$
$7 \times 3 - 12 = 9$
$8 \times 4 - 21 = 11$
$9 \times 5 - 32 = 13$
$. . . \times . . . - . . . = . . .$
$. . . \times . . . - . . . = . . .$
$. . . \times . . . - . . . = . . .$

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Question 10

2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

Accepted Answer

Let number of rows = xEach row contain plants = xTotal plants = 2025Therefore,$x^{2} = 2025$$x=45$.

Patterns in Whole Numbers

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QUICK SUMMARY WITH STORIES

Representing Whole Numbers By Dots In Different Geometrical Pattern

What is the total number of numbers less than $1000$ and divisible by $5$ , which can be formed with digits $0$ to $9$ , such that no digit gets repeated?

In a fraction, twice the numerator is $2$ more than the denominator.If $3$ is added to the numerator and to the denominator, the new fraction is $\frac{2}{3}$ .Find the original fraction.

Study the following pattern:
$1 + 3 = 2 \times 2$
$1 + 3 + 5 = 3 \times 3$
$1 + 3 + 7 = 4 \times 4$
$1 + 3 + 5 + 7 + 9 = 5 \times 5$
By observing the above pattern, find
(i) $1 + 3 + 5 + 7 + 9 + 11$
(ii) $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15$
(iii) $21 + 23 + 25 + + 51$

Write the first four products of two numbers with difference 4 starting from In the following order: 1, 2, 3, 4, 5, 6, .....
Identify the pattern in the products and write the next three products.

The value of question mark is :

Observe the pattern in the following and fill in the blanks:
$9 \times 9 + 7 = 88$
$98 \times 9 + 6 = 888$
$987 \times 9 + 5 = 8888$
$9876 \times 9 + 4 = . . . . . .$
$98765 \times 9 + 3 = . . . .$
$987654 \times 9 + 2 = . . . .$
$9876543 \times 9 + 1 = . . . . . .$

Study the following pattern:

$1 = \frac{1 \times 2}{2}$

$1 + 2 = \frac{2 \times 3}{2}$

$1 + 2 + 3 = \frac{3 \times 4}{2}$

$1 + 2 + 3 + 4 = \frac{4 \times 5}{2}$

By observing the above pattern, find
$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10$

If abc-cba = 792, where abc is a 3-digit number, if a+b+c = 18 and a = 9c, what is the number?

Observe the following pattern and extend it to three more steps:
$6 \times 2 - 5 = 7$
$7 \times 3 - 12 = 9$
$8 \times 4 - 21 = 11$
$9 \times 5 - 32 = 13$
$. . . \times . . . - . . . = . . .$
$. . . \times . . . - . . . = . . .$
$. . . \times . . . - . . . = . . .$

2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

Patterns in Whole Numbers

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Rate

QUICK SUMMARY WITH STORIES

Representing Whole Numbers By Dots In Different Geometrical Pattern

What is the total number of numbers less than 1000 and divisible by 5, which can be formed with digits 0 to 9, such that no digit gets repeated?

In a fraction, twice the numerator is 2 more than the denominator.If 3 is added to the numerator and to the denominator, the new fraction is 32​.Find the original fraction.

Study the following pattern: 1+3=2×2 1+3+5=3×3 1+3+7=4×4 1+3+5+7+9=5×5 By observing the above pattern, find (i) 1+3+5+7+9+11 (ii) 1+3+5+7+9+11+13+15 (iii) 21+23+25++51

Write the first four products of two numbers with difference 4 starting from In the following order: 1, 2, 3, 4, 5, 6, ..... Identify the pattern in the products and write the next three products.

The value of question mark is :

Observe the pattern in the following and fill in the blanks: 9×9+7=88 98×9+6=888 987×9+5=8888 9876×9+4=...... 98765×9+3=.... 987654×9+2=.... 9876543×9+1=......

Study the following pattern: 1=21×2​ 1+2=22×3​ 1+2+3=23×4​ 1+2+3+4=24×5​ By observing the above pattern, find 1+2+3+4+5+6+7+8+9+10

If abc-cba = 792, where abc is a 3-digit number, if a+b+c = 18 and a = 9c, what is the number?

Observe the following pattern and extend it to three more steps: 6×2−5=7 7×3−12=9 8×4−21=11 9×5−32=13 ...×...−...=... ...×...−...=... ...×...−...=...

2025 plants are to be planted in a garden in such a way that each row contains as many plants as the number of rows. Find the number of rows and the number of plants in each row.

What is the total number of numbers less than $1000$ and divisible by $5$ , which can be formed with digits $0$ to $9$ , such that no digit gets repeated?

In a fraction, twice the numerator is $2$ more than the denominator.If $3$ is added to the numerator and to the denominator, the new fraction is $\frac{2}{3}$ .Find the original fraction.

Study the following pattern:
$1 + 3 = 2 \times 2$
$1 + 3 + 5 = 3 \times 3$
$1 + 3 + 7 = 4 \times 4$
$1 + 3 + 5 + 7 + 9 = 5 \times 5$
By observing the above pattern, find
(i) $1 + 3 + 5 + 7 + 9 + 11$
(ii) $1 + 3 + 5 + 7 + 9 + 11 + 13 + 15$
(iii) $21 + 23 + 25 + + 51$

Write the first four products of two numbers with difference 4 starting from In the following order: 1, 2, 3, 4, 5, 6, .....
Identify the pattern in the products and write the next three products.

Observe the pattern in the following and fill in the blanks:
$9 \times 9 + 7 = 88$
$98 \times 9 + 6 = 888$
$987 \times 9 + 5 = 8888$
$9876 \times 9 + 4 = . . . . . .$
$98765 \times 9 + 3 = . . . .$
$987654 \times 9 + 2 = . . . .$
$9876543 \times 9 + 1 = . . . . . .$

Study the following pattern:

$1 = \frac{1 \times 2}{2}$

$1 + 2 = \frac{2 \times 3}{2}$

$1 + 2 + 3 = \frac{3 \times 4}{2}$

$1 + 2 + 3 + 4 = \frac{4 \times 5}{2}$

By observing the above pattern, find
$1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10$

Observe the following pattern and extend it to three more steps:
$6 \times 2 - 5 = 7$
$7 \times 3 - 12 = 9$
$8 \times 4 - 21 = 11$
$9 \times 5 - 32 = 13$
$. . . \times . . . - . . . = . . .$
$. . . \times . . . - . . . = . . .$
$. . . \times . . . - . . . = . . .$