Arithmetic aptitude :: Problems on Numbers :: General Questions

• Problems on Numbers - General Questions
• Problems on Numbers - Important Formulas and Facts

Basic Formulae:
1). $(a+b)(a-b)=(a^{2}-b^{2})$.
2). $(a+b)^{2}=(a^{2}+b^{2}+2ab)$.
3). $(a-b)^{2}=(a^2+b^{2}-2ab)$.
4). $(a+b+c)^{2} = a^{2}+b^{2}+c^{2}+2(ab+bc+ca)$.
5). $(a^{3}+b^{3})=(a+b)(a^{2}-ab+b^{2})$.
6). $(a^{3}-b^{3})= (a-b)(a^{2}+ab+b^{2})$.
7). $(a^{3}+b^{3}+c^{3}-3abc)=(a+b+c)(a^{2}+b^{2}+c^{2}-ab-bc-ac)$.
8). When $a+b+c = 0$, then $a^{3}+b^{3}+c^{3}=3abc$.

The sum of two positive integers multiplied by the bigger number is 204, and their difference multiplied by the smaller number is 35. The numbers are :

Answer:

Explanation:

Let the numbers be $x$ and $y$ such that $x>y$.Then,
$x(x+y)=204 \Rightarrow x^{2}+xy = 204 ----(1)$
and $y(x-y)=35 \Rightarrow xy -y^{2}= 35 ----(2)$.
Subtracting (2) from (1), we get : $x^{2}+y^{2}=169$.
The only triplet satisfying condition is $(12,5,13)$. Thus, $x=12$, $y=5$.

The difference between two numbers is 1365. When the larger number is divided by the smaller one, the quotient is 6 and the remainder is 15. The smaller number is :

Answer:

Explanation:

Let the numbers be $x$ and $(x+1365)$.
Then, $x+1365=6x+15 \Leftrightarrow 5x = 1350 \Leftrightarrow x = 270$.

The product of two numbers is 9375 and the quotient, when the larger one is divided by the smaller, is 15. The sum of the numbers is :

Answer:

Explanation:

Let the numbers be $x$ and $y$. Then, $xy=9375$ and $\frac{x}{y}=15$.
$\frac{xy}{(x/y)} =\frac{9375}{15} \Leftrightarrow y^{2} = 625$
$\Leftrightarrow y = 25 \Rightarrow x = 15y = (15 \times 25)= 375$.
$\therefore$ Sum of the numbers $= 375 + 25 = 400$.

Two numbers differ by 5. If their product is 336, then the sum of the two numbers is :

Answer:

Explanation:

Let the numbers be $x$ and $y$. Then, $x-y=5$ and $xy=336$.
$(x+y)^{2}=(x-y)^{2}+4xy$
$=25+4 \times 336 = 1369$
$\Rightarrow x+y = \sqrt{1369} = 37$.

The product of two numbers is 192 and the sum of these two numbers is 28. What is the smaller of these two numbers ?

Answer:

Explanation:

Let the numbers be $x$ and $(28-x)$. Then,
$x(28-x)=192 \Leftrightarrow x^{2}-28x+192=0$
$\Leftrightarrow (x-16)(x-12)=0$
$\Leftrightarrow x =16$ or $x=12$.

• Problems on Numbers - General Questions