Given, a x a x b x b x b x c x c x c x c
∵ a x a = a² and b x b x b = b³
and c x c x c x c = c⁴
So, a x a x b x b x b x c x c x c x c = a²b³c⁴

32bananas

Here, we want to establish that
ΔPQR ≅ ΔFED  [by SAS congruence rule]
Given that, PQ = FE and RP = DF
So, the additional information needed to establish the congruence is ㄥP = ㄥF. 

We have, 10p = 100
On dividing both sides by 10, we get
\(\frac{10p}{10}=\frac{100}{10}\quad \Rightarrow\) p = 10
Hence, p = 10 is the solution of the given equation.

We have: 5x +\(\frac{1}{3}\)=2-3x
\(\Rightarrow \) 5x+3x=2-\(\frac{1}{3}\)
[Transposing (-3x) to L.H.S. and 1. to R.H.S.]
\(\Rightarrow \) 8x=\(\frac{6-1}{3}=\frac{5}{3}\)
\(\Rightarrow \) x=\(\frac{5}{3}\times \frac{1}{8}=\frac{5}{24}\)
Thus, x=\(\frac{5}{24}\)
is the required solution of the given equation.

Given, 8.01 x 10⁷
∵ 10⁷ =10000000
and 8.01 = 801 x10^-2
So, 8.01 x 10⁷ = 801 x10^-2 x 10⁷ = 801 x 10⁵
= 801 x 100000 [∵ 10⁵ =100000]
= 80100000

We have, 75% of 12 = \(-{75\over100}\) x 12 = -3 x 12 = 3 x 3 = 9
Hence, 75% of 12 is 9.

In Δ ABC and Δ PQR, there are side possible matchings or correspondences. Out of them, four correspondences are as follow:
(i) ABC ↔️ PRQ
(ii) ABC ↔️ RPQ
(iii) ABC ↔️ RQP
(iv) ABC ↔️ QPR
Yes, all these correspondences may lead to congruence.

9.5853 x 10³

In some special cases (which depend on the lengths of the sides and the size of the angle involved), SSA is enough to show congruence. However, it is not always enough.
Consider the following triangles :

Here side AB is congruent to side DE (S) side AC is congruent to side DF (S) angle C is congruent to angle F (A)
But the triangles are not congruent, as we can see.
What happens is this : If we draw a vertical line through point A in the first triangle, we can sort of "flip" side AB around this line to get the second triangle, If we were to lay one triangle on top of the other and draw the vertical line, this how it would look.

Clearly, side DE is just side AB flipped around the line. So, we have not changed the length of the side, and the other side AC (or DF) is unchanged, as is angle C (or F). So, these two triangles that have the same SSA information, but they are not congruent.

\(\frac { 8 }{ 3 } \)

As per the given information in question,
cost price of per table = Rs 450
Number of tables = 24
So, cost price of 24 tables = 24 x 450 = Rs 10800
Selling price of per table = Rs 600
Number of tables sold at rate Rs  600 =16
Selling price of 16 tables = 16 x 600 =Rs 9600
\(\therefore\) Remaining tables = 24 -16 = 8
\(\because\) 8 tables sold at Rs 400.
Selling price for 8 tables = 8 x 400 = Rs 3200
Total selling price = 9600 + 3200 = Rs12800
\(\therefore\) Profit or Gain = Rs12800 - Rs10800 = Rs2000
Now, Gain. percentage=\({Gain \over Total \ cost \ price}\times 100\)
\(={2000\over10800}\times100={2000\over108}=18.51\%\)
Hence, his gain is 18.51%.

Let the number of two rupee coins be y.
Then the number of one rupee coins is 3y. Total money by two rupee coins = 2 x y = 2y
Total money by one rupee coin = 1 x 3y = 3y
Total worth of coins = Rs 120
So, the equation is 2y + 3y = 120  \(\Rightarrow\) 5y = 120
On dividing both sides by 5, we get
         \(\frac{5y}{5}=\frac{120}{5}\Rightarrow y=24\)
\(\therefore\) Number of two rupee coins = y = 24 and number of one rupee coins = 3y = 3 x 24 = 72.