Problem of the Week

Updated at Aug 31, 2020 5:40 PM

This week's problem comes from the equation category.

How would you solve the equation 54t×5t−3=2516\frac{5}{4t}\times \frac{5}{t-3}=\frac{25}{16}?

Let's begin!



54t×5t−3=2516\frac{5}{4t}\times \frac{5}{t-3}=\frac{25}{16}

1
Use this rule: ab×cd=acbd\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}.
5×54t(t−3)=2516\frac{5\times 5}{4t(t-3)}=\frac{25}{16}

2
Simplify  5×55\times 5  to  2525.
254t(t−3)=2516\frac{25}{4t(t-3)}=\frac{25}{16}

3
Multiply both sides by 4t(t−3)4t(t-3).
25=2516×4t(t−3)25=\frac{25}{16}\times 4t(t-3)

4
Use this rule: ab×cd=acbd\frac{a}{b} \times \frac{c}{d}=\frac{ac}{bd}.
25=25×4t(t−3)1625=\frac{25\times 4t(t-3)}{16}

5
Simplify  25×4t(t−3)25\times 4t(t-3)  to  100t(t−3)100t(t-3).
25=100t(t−3)1625=\frac{100t(t-3)}{16}

6
Simplify  100t(t−3)16\frac{100t(t-3)}{16}  to  25t(t−3)4\frac{25t(t-3)}{4}.
25=25t(t−3)425=\frac{25t(t-3)}{4}

7
Multiply both sides by 44.
100=25t(t−3)100=25t(t-3)

8
Expand.
100=25t2−75t100=25{t}^{2}-75t

9
Move all terms to one side.
100−25t2+75t=0100-25{t}^{2}+75t=0

10
Factor out the common term 2525.
25(4−t2+3t)=025(4-{t}^{2}+3t)=0

11
Factor out the negative sign.
25×−(t2−3t−4)=025\times -({t}^{2}-3t-4)=0

12
Divide both sides by 2525.
−t2+3t+4=0-{t}^{2}+3t+4=0

13
Multiply both sides by −1-1.
t2−3t−4=0{t}^{2}-3t-4=0

14
Factor t2−3t−4{t}^{2}-3t-4.
(t−4)(t+1)=0(t-4)(t+1)=0

15
Solve for tt.
t=4,−1t=4,-1

Done
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