Question paper, Set 10

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

$\displaystyle -\frac{1}{4}$, $16$ and $\displaystyle -\frac{3}{8}$ have fixed values, so they are constants.

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

Answer: $r$

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

Constants have a fixed value: $\displaystyle -\frac{8}{9}$, $25$ and $\displaystyle -\frac{7}{8}$.

Answer: (i) $q$, $m$, $a$ (ii) $\displaystyle -\frac{8}{9}$, $25$, $\displaystyle -\frac{7}{8}$

Take $22$ away from $69$.

$69 - 22 = 47$

Answer: $69 - 22 = 47$

(i) $36$ divided by $t$ is $\displaystyle \frac{36}{t}$

(ii) Take $15n$ away from $39$. $39 - 15n$

Answer: (i) $\displaystyle \frac{36}{t}$, (ii) $39 - 15n$

(i) Take $11q$ away from $32$. $32 - 11q$

(ii) $11$ times $t$ is $11t$; $10$ times $b$ is $10b$. Add $11t$ to $10b$: $10b + 11t$

(iii) $r$ divided by $12$ is $\displaystyle \frac{r}{12}$

(iv) $47$ more than $m$: add $47$ to $m$. $m + 47$

(v) Multiply: $13 \times c = 13c$

Answer: (i) $32 - 11q$, (ii) $10b + 11t$, (iii) $\displaystyle \frac{r}{12}$, (iv) $m + 47$, (v) $13c$

(i) One third of $x$ is $\displaystyle \frac{x}{3}$. $5$ less than it: take away $5$. $\displaystyle \frac{x}{3} - 5$

(ii) One third of $p$ is $\displaystyle \frac{p}{3}$. $5$ more than it: add $5$. $\displaystyle \frac{p}{3} + 5$

Answer: (i) $\displaystyle \frac{x}{3} - 5$, (ii) $\displaystyle \frac{p}{3} + 5$

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

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

Answer: $1$

(i) Rule $0 \times a = 0$, so $0 \times pr = 0$

(ii) Rule $-1 \times a = -a$, so $-1 \times b = -b$

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

Answer: (i) $0$, (ii) $-b$, (iii) $qr$

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

$9b - m$ and $a + 11$ use $+$ (or) $-$, so they are not terms.

Terms: $\displaystyle \frac{2}{r}$, $\displaystyle \frac{14}{z}$ and $\displaystyle \frac{13n}{r}$.

Answer: $\displaystyle \frac{2}{r}$, $\displaystyle \frac{14}{z}$, $\displaystyle \frac{13n}{r}$

(i) $9xz = 9 \times x \times z$, so the factors are $9, x, z$.

(ii) $-10mnp = -10 \times m \times n \times p$, so the factors are $-10, m, n, p$.

Answer: (i) $9, x, z$, (ii) $-10, m, n, p$

(i) $12a = 12 \times a$, so the coefficient is $12$.

(ii) $15ac = 15c \times a$, so the coefficient is $15c$.

(iii) $14am = 14m \times a$, so the coefficient is $14m$.

Answer: (i) $12$, (ii) $15c$, (iii) $14m$

$\displaystyle -\frac{qr}{10} = -\frac{1}{10} \times qr$

So the coefficient of $qr$ is $\displaystyle -\frac{1}{10}$.

Answer: $\displaystyle -\frac{1}{10}$

(i) $4pqr$, $-8pqr$: the same literal factors $pqr$, so like terms.

(ii) $\displaystyle -\frac{5}{4}mn$ has literal factors $mn$, $-9m$ has literal factors $m$: different, so unlike terms.

Answer: (i) Like, (ii) Unlike

Like terms have the same literal factors.

Literal factor $y$: $13y$, $11y$ and $3y$.

Literal factor $n$: $-12n$, $8n$ and $-7n$.

Literal factor $bc$: $4bc$ and $7bc$.

Answer: $13y$, $11y$, $3y$; $-12n$, $8n$, $-7n$; $4bc$, $7bc$

$30bc + 4bc = (30 + 4)bc$

$= 34bc$

Answer: $34bc$

(i) $4yz + 12yz = (4 + 12)yz$ $= 16yz$

(ii) $2mn + 9mn = (2 + 9)mn$ $= 11mn$

Answer: (i) $16yz$, (ii) $11mn$

Bring the like terms together and add.

$19pq + 3pq + 20mn + 2mn = (19pq + 3pq) + (20mn + 2mn)$

$= (19 + 3)pq + (20 + 2)mn$

$= 22pq + 22mn$

Answer: $22pq + 22mn$

(i) $-5xy + 23xy = (-5 + 23)xy = 18xy$

(ii) $8xy - 20xy = (8 - 20)xy = -12xy$

(iii) $-28xy - 11xy = (-28 - 11)xy = -39xy$

Answer: (i) $18xy$, (ii) $-12xy$, (iii) $-39xy$

(i) $-12ab - 28ab = (-12 - 28)ab = -40ab$

(ii) $-7ab + 5ab = (-7 + 5)ab = -2ab$

(iii) $24ab - 18ab = (24 - 18)ab = 6ab$

(iv) $-11qr + 25qr = (-11 + 25)qr = 14qr$

(v) $22qr - 24qr = (22 - 24)qr = -2qr$

Answer: (i) $-40ab$, (ii) $-2ab$, (iii) $6ab$, (iv) $14qr$, (v) $-2qr$

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

$9ab + 8ac - 17m + 15ab - 20ac - 9m - 20ab = (9ab + 15ab - 20ab) + (8ac - 20ac) + (-17m - 9m)$

$= (9 + 15 - 20)ab + (8 - 20)ac + (-17 - 9)m$

$= 4ab - 12ac - 26m$

Answer: $4ab - 12ac - 26m$