12. Find the initial amount if the final amount, after an increase of: $$ \begin{array}{ll} ext { (a) } 20 \% ext { was equal to } \mathrm{R} 84 & ext { (b) } 25 \% ext { was equal to } 150 \mathrm{~kg} \ ext { (c) } 10 \% ext { was equal to } 55 \mathrm{~km} & ext { (d) } 40 \% ext { was equal to } 560 \mathrm{~g} \ ext { (e) } \frac{1}{5} ext { was equal to } 240 \mathrm{~km} & ext { (f) } \frac{1}{8} ext { was equal to R } 180\end{array} $$ 13. Find the initial amount if the final amount, after a decrease of: $$ \begin{array}{ll} ext { (a) } 20 \% ext { was equal to R72 } & ext { (b) } 25 \% ext { was equal to } 120 \mathrm{~kg} \ ext { (c) } 10 \% ext { was equal to } 45 \mathrm{~km} & ext { (d) } 40 \% ext { was equal to } 360 \mathrm{~g} \ ext { (e) } \frac{1}{4} ext { was equal to R300 } & ext { (f) } \frac{1}{6} ext { was equal to } 80 \mathrm{~m}\end{array} $$

Question

12. Find the initial amount if the final amount, after an increase of: $$ \begin{array}{ll}	ext { (a) } 20 \% 	ext { was equal to } \mathrm{R} 84 & 	ext { (b) } 25 \% 	ext { was equal to } 150 \mathrm{~kg} \ 	ext { (c) } 10 \% 	ext { was equal to } 55 \mathrm{~km} & 	ext { (d) } 40 \% 	ext { was equal to } 560 \mathrm{~g} \ 	ext { (e) } \frac{1}{5} 	ext { was equal to } 240 \mathrm{~km} & 	ext { (f) } \frac{1}{8} 	ext { was equal to R } 180\end{array} $$ 13. Find the initial amount if the final amount, after a decrease of: $$ \begin{array}{ll}	ext { (a) } 20 \% 	ext { was equal to R72 } & 	ext { (b) } 25 \% 	ext { was equal to } 120 \mathrm{~kg} \ 	ext { (c) } 10 \% 	ext { was equal to } 45 \mathrm{~km} & 	ext { (d) } 40 \% 	ext { was equal to } 360 \mathrm{~g} \ 	ext { (e) } \frac{1}{4} 	ext { was equal to R300 } & 	ext { (f) } \frac{1}{6} 	ext { was equal to } 80 \mathrm{~m}\end{array} $$

UpStudy AI · Accepted Answer

**12. Final amount after an increase**

Let the initial amount be $$ I $$. An increase of a fraction $$ r $$ means the final amount is 
$$
I(1+r).
$$

**(a)** Increase of $$20\%$$ means $$ r=0.20 $$:
$$
1.2I = 84 \quad \Rightarrow \quad I = \frac{84}{1.2} = 70.
$$

**(b)** Increase of $$25\%$$ means $$ r=0.25 $$:
$$
1.25I = 150 \quad \Rightarrow \quad I = \frac{150}{1.25} = 120.
$$

**(c)** Increase of $$10\%$$ means $$ r=0.10 $$:
$$
1.1I = 55 \quad \Rightarrow \quad I = \frac{55}{1.1} = 50.
$$

**(d)** Increase of $$40\%$$ means $$ r=0.40 $$:
$$
1.4I = 560 \quad \Rightarrow \quad I = \frac{560}{1.4} = 400.
$$

**(e)** Increase of $$\frac{1}{5}$$ (i.e. $$20\%$$) means $$ r=\frac{1}{5} $$:
$$
\left(1+\frac{1}{5}\right)I = 240 \quad \Rightarrow \quad 1.2I = 240 \quad \Rightarrow \quad I = \frac{240}{1.2} = 200.
$$

**(f)** Increase of $$\frac{1}{8}$$ (i.e. $$12.5\%$$) means $$ r=\frac{1}{8} $$:
$$
\left(1+\frac{1}{8}\right)I = 180 \quad \Rightarrow \quad 1.125I = 180 \quad \Rightarrow \quad I = \frac{180}{1.125} = 160.
$$

---

**13. Final amount after a decrease**

For a decrease of a fraction $$ d $$, the final amount is 
$$
I(1-d).
$$

**(a)** Decrease of $$20\%$$ means $$ d=0.20 $$:
$$
0.8I = 72 \quad \Rightarrow \quad I = \frac{72}{0.8} = 90.
$$

**(b)** Decrease of $$25\%$$ means $$ d=0.25 $$:
$$
0.75I = 120 \quad \Rightarrow \quad I = \frac{120}{0.75} = 160.
$$

**(c)** Decrease of $$10\%$$ means $$ d=0.10 $$:
$$
0.9I = 45 \quad \Rightarrow \quad I = \frac{45}{0.9} = 50.
$$

**(d)** Decrease of $$40\%$$ means $$ d=0.40 $$:
$$
0.6I = 360 \quad \Rightarrow \quad I = \frac{360}{0.6} = 600.
$$

**(e)** Decrease of $$\frac{1}{4}$$ (i.e. $$25\%$$) means $$ d=\frac{1}{4} $$:
$$
0.75I = 300 \quad \Rightarrow \quad I = \frac{300}{0.75} = 400.
$$

**(f)** Decrease of $$\frac{1}{6}$$ (approximately $$16.67\%$$) means $$ d=\frac{1}{6} $$:
$$
\frac{5}{6}I = 80 \quad \Rightarrow \quad I = 80 \times \frac{6}{5} = 96.
$$
**12. Initial Amount After an Increase** - **(a)** Initial amount: R70 - **(b)** Initial amount: 120 kg - **(c)** Initial amount: 50 km - **(d)** Initial amount: 400 g - **(e)** Initial amount: 200 km - **(f)** Initial amount: 160 --- **13. Initial Amount After a Decrease** - **(a)** Initial amount: R90 - **(b)** Initial amount: 160 kg - **(c)** Initial amount: 50 km - **(d)** Initial amount: 600 g - **(e)** Initial amount: 400 - **(f)** Initial amount: 96

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