Fundamental Concepts of Algebra — Exercise 12(A)

Write the following in mathematical form using signs and symbols:

$x$ increased by 23.

Answer

$x$ increased by 23 means 23 is added to $x$.

Hence, the required form is $\bm{x + 23}$.

Write the following in mathematical form using signs and symbols:

$y$ decreased by 15.

Answer

$y$ decreased by 15 means 15 is subtracted from $y$.

Hence, the required form is $\bm{y - 15}$.

Write the following in mathematical form using signs and symbols:

$x$ minus thrice $y$.

Answer

Thrice $y = 3y$.

Hence, the required form is $\bm{x - 3y}$.

Write the following in mathematical form using signs and symbols:

$x$ taken away from 20.

Answer

$x$ taken away from 20 means $x$ is subtracted from 20.

Hence, the required form is $\bm{20 - x}$.

Write the following in mathematical form using signs and symbols:

Four times $x$ decreased by five times $y$.

Answer

Four times $x = 4x$ and five times $y = 5y$.

Hence, the required form is $\bm{4x - 5y}$.

Write the following in mathematical form using signs and symbols:

Sum of $x$ and quotient of $y$ by 10.

Answer

Quotient of $y$ by $10 = \dfrac{y}{10}$.

Hence, the required form is $\bm{x + \dfrac{y}{10}}$.

Write the following in mathematical form using signs and symbols:

Product of $a$ and $b$ subtracted from their sum.

Answer

Sum of $a$ and $b = a + b$, and product of $a$ and $b = ab$.

Hence, the required form is $\bm{(a + b) - ab}$.

Write the following in mathematical form using signs and symbols:

Quotient of $a$ by 3, multiplied by the sum of $b$ and 3.

Answer

Quotient of $a$ by $3 = \dfrac{a}{3}$ and sum of $b$ and $3 = (b + 3)$.

Hence, the required form is $\bm{\dfrac{a}{3}(b + 3)}$.

Write the following statement in mathematical form using signs and symbols:

$x$ exceeds 14 by 5.

Answer

$x$ exceeds 14 by 5 means $x - 14 = 5$.

Hence, the required form is $\bm{x - 14 = 5}$.

Write the following statement in mathematical form using signs and symbols:

Twice $x$ decreased by 9 gives 11.

Answer

Twice $x = 2x$. Decreased by 9 gives $2x - 9$, and this equals 11.

Hence, the required form is $\bm{2x - 9 = 11}$.

Write the following statement in mathematical form using signs and symbols:

45 decreased by $y$ gives 29.

Answer

45 decreased by $y = 45 - y$, and this equals 29.

Hence, the required form is $\bm{45 - y = 29}$.

Write the following statement in mathematical form using signs and symbols:

One third of the sum of $x$ and 4 equals 10.

Answer

Sum of $x$ and $4 = (x + 4)$. One third of it $= \dfrac{1}{3}(x + 4)$, which equals 10.

Hence, the required form is $\bm{\dfrac{1}{3}(x + 4) = 10}$.

Write the following statement in mathematical form using signs and symbols:

16 less than the quotient of $a$ by 2 equals 8.

Answer

Quotient of $a$ by 2 = $\dfrac{a}{2}$. 16 less than it = $\dfrac{a}{2} - 16$, which equals 8.

Hence, the required form is $\bm{\dfrac{a}{2} - 16 = 8}$.

Write the following statement in mathematical form using signs and symbols:

Nine times $m$ is greater than 18.

Answer

Nine times $m = 9m$, which is greater than 18.

Hence, the required form is $\bm{9m \gt 18}$.

Write the following statement in mathematical form using signs and symbols:

7 is greater than twice $p$.

Answer

Twice $p = 2p$, and 7 is greater than it.

Hence, the required form is $\bm{7 \gt 2p}$.

Write the following statement in mathematical form using signs and symbols:

The quotient of $a$ by 6 is less than 1.

Answer

Quotient of $a$ by $6 = \dfrac{a}{6}$, which is less than 1.

Hence, the required form is $\bm{\dfrac{a}{6} \lt 1}$.

Write the following statement in mathematical form using signs and symbols:

Three times $x$ is 12 less than twice $y$.

Answer

Three times $x = 3x$ and twice $y = 2y$. Since $3x$ is 12 less than $2y$, we get $3x = 2y - 12$.

Hence, the required form is $\bm{3x = 2y - 12}$.

Write the following in exponential form:

$a \times a \times a \times .....$ taken 10 times.

Answer

When the same factor is multiplied repeatedly, we write it using exponents (powers).

$a \times a \times a \times .....$ taken 10 times = $a^{10}$.

Hence, the exponential form is $\bm{a^{10}}$.

Write the following in exponential form:

$r \times r \times r \times .....$ taken 6 times.

Answer

When the same factor is multiplied repeatedly, we write it using exponents (powers).

$r \times r \times r \times .....$ taken 6 times = $r^{6}$

Hence, the exponential form is $\bm{r^6}$.

Write the following in exponential form:

$2 \times m \times m \times n \times n \times n$.

Answer

When the same factor is multiplied repeatedly, we write it using exponents (powers).

$\Rightarrow 2 \times m \times m \times n \times n \times n\\[1em] \Rightarrow 2 \times m^2 \times n^3\\[1em] \Rightarrow \bm{2m^2n^3}$

Hence, the exponential form is $\bm{2m^2n^3}$.

Write the following in exponential form:

$3 \times 5 \times p \times p \times p \times q \times q \times r$.

Answer

When the same factor is multiplied repeatedly, we write it using exponents (powers).

$\Rightarrow 3 \times 5 \times p \times p \times p \times q \times q \times r\\[1em] \Rightarrow 15 \times p^3 \times q^2 \times r\\[1em] \Rightarrow \bm{15p^3q^2r}$

Hence, the exponential form is $\bm{15p^3q^2r}$.

Write the following in product form:

$6x^2y^3$

Answer

Product form means expanding exponents into repeated multiplication.

Hence, the product form is $\bm{6 \times x \times x \times y \times y \times y}$.

Write the following in product form:

$3a^2b^3c^4$

Answer

Product form means expanding exponents into repeated multiplication.

Hence, the product form is $\bm{3 \times a \times a \times b \times b \times b \times c \times c \times c \times c}$.

Write the following in product form:

$2p^3q^3r^3$

Answer

Product form means expanding exponents into repeated multiplication.

Hence, the product form is $\bm{2 \times p \times p \times p \times q \times q \times q \times r \times r \times r}$.