Question paper, Set 8

A variable (or literal) can take various numerical values. A constant has a fixed value.

$15$, $-22$ and $6$ have fixed values, so they are constants.

$x$ can take various values, so it is the variable.

Answer: $x$

Variables can take various numerical values: $p$, $m$ and $q$.

Constants have a fixed value: $-10$, $-14$ and $-7$.

Answer: (i) $p$, $m$, $q$ (ii) $-10$, $-14$, $-7$

Take $t$ away from $y$.

$y - t$

Answer: $y - t$

(i) Take $56$ away from $b$. $b - 56$

(ii) $33$ divided by $n$ is $\displaystyle \frac{33}{n}$

Answer: (i) $b - 56$, (ii) $\displaystyle \frac{33}{n}$

(i) $2$ times $n$ is $2n$; $9$ times $m$ is $9m$. Add $2n$ to $9m$: $9m + 2n$

(ii) $31$ divided by $z$ is $\displaystyle \frac{31}{z}$

(iii) Take $7a$ away from $16$. $16 - 7a$

(iv) Take $39$ away from $m$. $m - 39$

(v) Multiply: $4 \times y = 4y$

Answer: (i) $9m + 2n$, (ii) $\displaystyle \frac{31}{z}$, (iii) $16 - 7a$, (iv) $m - 39$, (v) $4y$

(i) Half of $q$ is $\displaystyle \frac{q}{2}$. $7$ less than it: take away $7$. $\displaystyle \frac{q}{2} - 7$

(ii) One third of $c$ is $\displaystyle \frac{c}{3}$. $14$ less than it: take away $14$. $\displaystyle \frac{c}{3} - 14$

Answer: (i) $\displaystyle \frac{q}{2} - 7$, (ii) $\displaystyle \frac{c}{3} - 14$

Rule: $\displaystyle \frac{a}{-1} = -a$

So $\displaystyle \frac{c}{-1} = -c$: the box is $-1$.

Answer: $-1$

(i) Rule $1 \times a = a$, so $1 \times yz = yz$

(ii) Rule $0 \times a = 0$, so $0 \times z = 0$

(iii) Rule $\displaystyle \frac{a}{1} = a$, so $\displaystyle \frac{r}{1} = r$

Answer: (i) $yz$, (ii) $0$, (iii) $r$

A term is made from constants and variables by multiplication (or) division only.

$m + r$ and $b + 14$ use $+$ (or) $-$, so they are not terms.

Terms: $\displaystyle \frac{9}{c}$, $13qr$ and $-28$.

Answer: $\displaystyle \frac{9}{c}$, $13qr$, $-28$

$45 = 1 \times 45$

$45 = 3 \times 15$

$45 = 5 \times 9$

So the factors of $45$ are $1$, $3$, $5$, $9$, $15$ and $45$.

Answer: $45 = 1 \times 45 = 3 \times 15 = 5 \times 9$

(i) $4n = 4 \times n$, so the coefficient is $4$.

(ii) $3np = 3p \times n$, so the coefficient is $3p$.

(iii) $-6ny = -6y \times n$, so the coefficient is $-6y$.

Answer: (i) $4$, (ii) $3p$, (iii) $-6y$

$\displaystyle -\frac{np}{5} = -\frac{1}{5} \times np$

So the coefficient of $np$ is $\displaystyle -\frac{1}{5}$.

Answer: $\displaystyle -\frac{1}{5}$

(i) $\displaystyle \frac{3}{2}x$ has literal factors $x$, $\displaystyle \frac{3}{2}q$ has literal factors $q$: different, so unlike terms.

(ii) $4pqt$, $-13pqt$: the same literal factors $pqt$, so like terms.

Answer: (i) Unlike, (ii) Like

Like terms have the same literal factors.

Literal factor $bc$: $13bc$ and $-11bc$.

Literal factor $mn$: $-9mn$, $6mn$ and $-4mn$.

Literal factor $b$: $-9b$, $-10b$ and $4b$.

Answer: $13bc$, $-11bc$; $-9mn$, $6mn$, $-4mn$; $-9b$, $-10b$, $4b$

$14x + 7x = (14 + 7)x$

$= 21x$

Answer: $21x$

(i) $25ac + 10ac = (25 + 10)ac$ $= 35ac$

(ii) $25z + 29z = (25 + 29)z$ $= 54z$

Answer: (i) $35ac$, (ii) $54z$

Letter terms together, numbers together.

$15xy + 4 + 14xy + 10 = (15xy + 14xy) + (4 + 10)$

$= (15 + 14)xy + (4 + 10)$

$= 29xy + 14$

Answer: $29xy + 14$

(i) $5np - 29np = (5 - 29)np = -24np$

(ii) $-27np - 10np = (-27 - 10)np = -37np$

(iii) $-22np + 6np = (-22 + 6)np = -16np$

Answer: (i) $-24np$, (ii) $-37np$, (iii) $-16np$

(i) $-16n - 28n = (-16 - 28)n = -44n$

(ii) $26n - 18n = (26 - 18)n = 8n$

(iii) $-4n + 6n = (-4 + 6)n = 2n$

(iv) $-22a + 12a = (-22 + 12)a = -10a$

(v) $23a - 27a = (23 - 27)a = -4a$

Answer: (i) $-44n$, (ii) $8n$, (iii) $2n$, (iv) $-10a$, (v) $-4a$

Bring the like terms together, then add (or) subtract the numbers.

$14xy - 7bc - 2mp - 9xy - 2mp - 17bc + 5bc = (14xy - 9xy) + (-7bc - 17bc + 5bc) + (-2mp - 2mp)$

$= (14 - 9)xy + (-7 - 17 + 5)bc + (-2 - 2)mp$

$= 5xy - 19bc - 4mp$

Answer: $5xy - 19bc - 4mp$