a) Recall that to convert a percentage into a fraction, we write the percentage as a fraction with $100$ on the denominator and write this fraction in its simplest form. Using this method, we have:

$$\var{a}\%=\simplify{{{a}}/{100}}$$

b) Using the same method as in part a), we have:

$$\var{b}\%=\simplify{{{b}}/{100}}$$

c) Using the same method as in parts a) and b), we have:

$$\var{c}\%=\simplify{{{c}}/{100}}$$

a) Method 1: Dividing $\var{a}$ by $\var{b}$ gives:

\[\simplify[!basic]{{a}/{b}}=\simplify{{{a}/{b}}}\]

Multiplying by $100$ gives us the required percentage: 

\[\simplify{100}\times \simplify{{{a}/{b}}}=\var{latex(dpformat(a_ans,2))}\text{ (to }2\text{d.p.)}\]

so 

\[\simplify[!basic]{{a}/{b}}=\var{latex(dpformat(a_ans,2))}\%\]

b) Method 1: Dividing $\var{b}$ by $\var{a}$ gives:

\[\simplify[!basic]{{b}/{a}}=\simplify{{{b}/{a}}}\]

Multiplying by $100$ gives us the required percentage: 

\[\simplify{100}\times \simplify{{{b}/{a}}}=\var{latex(dpformat(b_ans,2))}\text{ (to }2\text{d.p.)}\]

so 

\[\simplify[!basic]{{b}/{a}}=\var{latex(dpformat(b_ans,2))}\%\]

c) Method 1: Dividing $\var{c}$ by $\var{d}$ gives:

\[\simplify[!basic]{{c}/{d}}=\simplify{{{c}/{d}}}\]

Multiplying by $100$ gives us the required percentage: 

\[\simplify{100}\times \simplify{{{c}/{d}}}=\var{latex(dpformat(c_ans,2))}\text{ (to }2\text{d.p.)}\]

so 

\[\simplify[!basic]{{c}/{d}}=\var{latex(dpformat(c_ans,2))}\%\]

d) Method 1: Dividing $\var{d}$ by $\var{c}$ gives:

\[\simplify[!basic]{{d}/{c}}=\simplify{{{d}/{c}}}\]

Multiplying by $100$ gives us the required percentage: 

\[\simplify{100}\times \simplify{{{d}/{c}}}=\var{latex(dpformat(d_ans,2))}\text{ (to }2\text{d.p.)}\]

so 

\[\simplify[!basic]{{d}/{c}}=\var{latex(dpformat(d_ans,2))}\%\]

a) Recall that to find $X\%$ of $Y$, the calculation is:

\[\dfrac{X}{100}\times Y\]

Here we have $X=\var{a}$ and $Y=\var{b}$, so $\var{a}\%$ of $\var{b}$ is:

\[\simplify[!simplifyFractions]{{{a}}/100}\times\var{b}=\var{latex(dpformat(a_ans,2))}\]

b) Here we have $X=\var{c}$ and $Y=\var{d}$, so $\var{c}\%$ of $\var{d}$ is:

\[\simplify[!simplifyFractions]{{{c}}/100}\times\var{d}=\var{latex(dpformat(b_ans,2))}\]

c) Here we have $X=\var{g}$ and $Y=\var{f}$, so $\var{g}\%$ of $\var{f}$ is:

\[\simplify[!simplifyFractions]{{{g}}/100}\times\var{f}=\var{latex(dpformat(c_ans,2))}\]

The formula for a percentage change is:

\[\dfrac{v_1-v_0}{v_0}\times 100\]

Here we have $v_0={\var{v_0}}$ and $v_1={\var{v_1}}$ so the percentage change is:

\[\dfrac{\var{v1}-\var{v0}}{\var{v0}}\times 100=\var{a_ans} \text{ (to $2$.d.p)}\]

So the percentage change to $2$.d.p is $\var{a_ans}\%$.

The formula for a percentage change is:

\[\dfrac{v_1-v_0}{v_0}\times 100\]

Here we have $v_0={\var{v_0}}$ and $v_1={\var{v_1}}$ so the percentage change is:

\[\dfrac{\var{v1}-\var{v0}}{\var{v0}}\times 100=\var{a_ans} \text{ (to $2$.d.p)}\]

So the percentage change to $2$.d.p is $\var{a_ans}\%$.

a) We know that Foodco produces $\frac{1}{\var{b}}$ of the combined output of the two firms, and that the combined output is $\var{a}\%$ of the town’s food industry output. To work out the percentage of the town’s food industry output Foodco produces, the first step is to find $\frac{1}{\var{b}}$ of $\var{a}\%$. Recall that we can write $\var{a}\%$ as $\frac{\var{a}}{100}$. To find $\frac{1}{\var{b}}$ of $\var{a}\%$, we must therefore multiply $\frac{1}{\var{b}}$ by $\frac{\var{a}}{100}$:

\[\frac{1}{\var{b}}\times \frac{\var{a}}{100}=\frac{\var{a}}{\simplify{{b}*100}}\]

The next step is to convert $\frac{\var{a}}{\simplify{{b}*100}}$ into a percentage:

Evaluating the fraction gives:

\[\frac{\var{a}}{\simplify{{b}*100}}=\simplify{{{a}/({b}*100)}}\]

Multiplying this by $100$ gives us the required percentage:

\[\simplify{{(1/{b})*{a}}/100}\times \var{a_ans}\%\]

So $\frac{1}{\var{b}}$ of $\var{a}\%$ is $\var{a_ans}\%$. Foodco’s produce is $\var{a_ans}\%$ of the town’s food industry output.

b) We know that Groceree produces $\var{d}\%$ of the combined output of the two firms, and that the combined output is $\var{a}\%$ of the town’s food industry output. To work out the fraction of the town’s food industry output Groceree produces we must find $\var{d}\%$ of $\var{a}\%$. We can write $\var{d}\%$ as $\frac{\var{d}}{100}$ and $\var{a}\%$ as $\frac{\var{a}}{100}$. To find $\var{d}\%$ of $\var{a}\%$ we must therefore multiply $\frac{\var{d}}{100}$ by $\frac{\var{a}}{100}$:

\[\frac{\var{d}}{100}\times \frac{\var{a}}{100}=\simplify{{{a}*{d}}/10000}\]

Since this is already a fraction, we have answered the question: Groceree’s produce is $\simplify{{{a}*{d}}/10000}$ of the town’s food industry output.

a) We want to express $\var{b}$ as a percentage of $\var{a}$. To do this, we must divide $\var{b}$ by $\var{a}$ to produce a fraction:

\[\frac{\var{b}}{\var{a}}\]

and then convert this fraction into a percentage by evaluating the fraction and multiplying by $100$:

\[\frac{\var{b}}{\var{a}}=\simplify{{{b}/{a}}}\]

\[100\times \simplify{{{b}/{a}}}=\var{latex(dpformat(ans_a,2))}\%\text{ (to $2$.d.p)}\]

b) Here we want to express $\var{c}$ as a percentage of $\var{d}$. First, we must divide $\var{c}$ by $\var{d}$ to produce a fraction:

\[\frac{\var{c}}{\var{d}}\]

We must then convert this fraction into a percentage by evaluating the fraction and multiplying by $100$:

\[\frac{\var{c}}{\var{d}}=\simplify{{{c}/{d}}}\]

\[100\times \simplify{{{c}/{d}}}=\var{latex(dpformat(ans_b,2))}\%\text{ (to $2$.d.p)}\]

c) We know that there are a total of $\var{d}$ machines and that $\var{f}\%$ of these broke down. We want to find $\var{f}\%$ of $\var{d}$. To do this, we divide $\var{f}$ by $100$ and multiply the result by $\var{d}$:

\[\frac{\var{f}}{100}\times \var{d}=\var{ans_b2}\]

d) The formula for a percentage change is:

\[\dfrac{v_1-v_0}{v_0}\times 100\]

Here we have $v_0={\var{g}}$ and $v_1={\var{h}}$ so the percentage change is:

\[\dfrac{\var{h}-\var{g}}{\var{g}}\times 100=\var{latex(dpformat(pchange,2))} \text{ (to $2$.d.p)}\]

e) Here we know that $\var{k}$ is equal to $\var{j}\%$ of the total number of chocolate eggs produced by the factory last Easter. To find the original (total) amount, we must multiply $\var{k}$ by $100$ and divide by $\var{j}$:

\[\dfrac{100\times \var{k}}{\var{j}}=\var{r}\]

So $\var{r}$ chocolate Easter eggs were produced by the factory last Easter.

f) By next summer, the workforce will have increased by $\var{l}\%$, or equivalently from $\var{m}$ to some new value. Let's call this new value $x$. To work out the value of $x$ we must first notice that $x$ is equal to $(100+\var{l})\%=\simplify{{100+{l}}}\%$ of $\var{m}$. In other words,

\[x=\dfrac{\simplify{{100+{l}}}}{100}\times \var{m}\]

\[=\var{ans_e}\]

So $\var{ans_e}$ people will be employed at the factory by next summer.

g) The formula for the PED is:
\[\textbf{PED }= \frac{\text{percentage change in quantity demanded}}{\text{percentage change in price}}\]

Using the formula for percentage changes, the percentage change in quantity demanded is:

\[100\times\frac{\var{high}-\var{low}}{\var{low}}=100\times\simplify{{{high}-{low}}/{low}}=\var{latex(dpformat(p_change_demand,2))}\%\text{ (to $2$.d.p)}\]

and the percentage change in price is:

\[100\times\frac{\var{latex(dpformat(p_low,2))}-\var{latex(dpformat(p_high,2))}}{\var{latex(dpformat(p_high,2))}}=100\times\simplify{{{p_low}-{p_high}}/{p_high}}=\var{latex(dpformat(p_change_price,2))}\%\text{ (to $2$.d.p)}\]

Now, using the formula for the PED we have:
\[\text{PED }=\frac{\var{latex(dpformat(p_change_demand,2))}}{\var{latex(dpformat(p_change_price,2))}}=\var{latex(dpformat(ans_g,2))}\text{ (to $2$.d.p)}\]

So the PED for the company's sheep food is $\var{ans_g2}$ (since we ignore the sign).