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12 Days of Christmas GMAT Competition 2025 - Day 2: If (3.7 * 10^a)

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Bunuel

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If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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Bunuel

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Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

GMAT Club Official Explanation:

$(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c $

$(3.7 * 5.8) * (10^a * 10^b) = 0.2146 * 10^c$

$21.46 * 10^{(a + b)} = 0.2146 * 10^c$

$(0.2146 * 10^2) * 10^{(a + b)} = 0.2146 * 10^c$

$0.2146 * 10^{(a + b + 2)} = 0.2146 * 10^c$

$10^{(a + b + 2)} = 10^c$

$c = a + b + 2$

Answer: B.

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3.7*5.8*10^a*10^b = 0.2146*10^c
21.46*10^a*10*b = 0.2146*10^c
10^a+b+2 = 10^c
Hence, a+b+2 = c

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Solving LHS
3.7 *5.8*10^(a+b)
= 21.46 * 10^(a+b)
= 0.2146 * 10^(a+b+2)

For LHS = RHS, c = a+b+2

Answer B

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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simplify it
0.37 * 0.58 * 10 ^(a + 1 + b + 1) = 0.2146 * 10^ ( a+ b + 2)
hence c = a + b +2

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(3.7*5.8)*10^(a+b) = 0.2146* 10^c
(21.46)*10^(a+b)=0.2146*10^c
(0.2146)*10^2 *10^(a+b) = 0.2146*10^c
Equating powers of 10 on both side, 2+a+b= c

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3.7 * 10^a * 5.8 * 10^ b = 21.46 * 10^{a+b} = 0.2146 * 10^2 * 10^ {a+b} = 0.2146 * 10^{a+b+2}

Option B

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The left side of the equation can be rewritten as:
3.7 * 10^a * 5.8 * 10^b = 3.7 * 5.8 * 10^(a+b) = 21.46*10^(a+b)

21.46 = 0.2146 * 10^2

Therefore the term 0.2146 needs to be multiplied by 100 to become 21.46

21.46 * 10^(a+b) = 21.46 * 10^-2 * 10^c
Divide both sides by 21.46 -> 10^(a+b) = 10^(c-2) -> a+b=c-2 -> c=a+b+2

Answer B

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$(3.7*10^a)*(5.8*10^b) = 0.2146*10^c$

$3.7*5.8 = 21.46 = 0.2146*10^2$

$0.2146*10^2*10^a*10^b = 0.2146*10^(2+a+b) = 0.2146*10^c$

Comparing power of 10 on both sides, we get

c = a + b + 2

IMO B

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(3.7*10^a)(5.8*10^b)=0.2146*10^c
37*58*10^(a+b-2) = 2146*10^(c-4)
2146*10^(a+b-2)= 2146*10^(c-4)
a+b-2=c-4
c=a+b-2+4
c= a+b+2

B

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If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

(3.7*10^a)* (5.8*10^b) = .2146*10^c
what is c in terms of a& b
(3.7*5.8)*(10^a+b) = .2146*10^c
21.46*10^a+b = .2146*10^c
.2146*10^2 * 10^a+b= .2146*10^c

2+a+b=c
option B is correct

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Step 1: 3.7 × 5.8 = 21.46
So it becomes 21.46 × 10^a+b

Then given 0.2146 × 10^c = 0.2146× 10^a+b+2

Answer is equate powers C = a+b+2

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Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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Deconstructing the Question
Given the equation:
$(3.7 \times 10^a) \times (5.8 \times 10^b) = 0.2146 \times 10^c$
Target: Find $c$ in terms of $a$ and $b$.

Step 1: Simplify the Left Hand Side (LHS)
Group the numerical coefficients and the powers of 10 separately:
$\text{LHS} = (3.7 \times 5.8) \times (10^a \times 10^b)$

Calculate the product of the numbers:
$3.7 \times 5.8 = 21.46$

Apply exponent rules ($x^m \cdot x^n = x^{m+n}$):
$10^a \times 10^b = 10^{a+b}$

So, the equation becomes:
$21.46 \times 10^{a+b} = 0.2146 \times 10^c$

Step 2: Align the Coefficients
To compare the exponents of 10, the decimal numbers on both sides must match.
We need to convert $21.46$ on the left to match $0.2146$ on the right.

Shift the decimal point 2 places to the left:
$21.46 = 0.2146 \times 10^2$

Substitute this back into the LHS:
$(0.2146 \times 10^2) \times 10^{a+b} = 0.2146 \times 10^c$

Combine the powers of 10 on the left:
$0.2146 \times 10^{2 + a + b} = 0.2146 \times 10^c$

Step 3: Equate the Exponents
Since the coefficients ($0.2146$) are now identical, the powers of 10 must be equal:
$10^{a + b + 2} = 10^c$

Therefore: $c = a + b + 2$

Answer: B

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Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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(3.7 * 10^a) * ( 5.8 * 10^b) = 0.2146 * 10^c

We need to express c in terms of an and b.

(37/10) * 10^a * (58 /10)*10^b = (2146/10000) * 10^c

We know that: (a^m) / (a^n) = a^(m-n)

37* 10^(a-1) * 58 * (10^b-1) = 2146 * 10^(c-4)

27*58 = 2146, thus the equation becomes

10^(a-1) * 10^(b-1) = 10^(c-4)

a^ m * a ^n = a^(m+n)

10^(a-1+b-1) = 10^(c-4)

Base being equal, we can equate the powers.

(a+b-2) = c-4

C = a+b+2

Option B

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Choose B
Explaination as below

Attachments

File comment: Choose B
Explanation as below

IMG_5284.jpg [ 1.9 MiB | Viewed 1887 times ]

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First multiply the numbers, combine the powers of 10, the rewrite 21.46 i the required form and you get the answer as c=a+b+2

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21.46*10^(a+b)=.2146^c;
.21.46*10^(a+b-2)=.2146*10^c

c=a+b-2

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Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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3.7 * 5.8 = 21.46

3.7 * 10^a * 5.8 * 10^b = 21.46 * 10^(a+b)

Multiplying and dividing by 100

We get 10^2 * 10^(a+b) * 21.64/100 = 10^(a+b+2) * 0.2146

Comparing LHS

a + b + 2 = c

Option B

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you can rearrange

(3.7 *5.8) (10^b*10^a) = 0.2146 *10^c

(0.2146 *10^2)

a+b + 2

Bunuel

If $(3.7 * 10^a) * (5.8 * 10^b) = 0.2146 * 10^c$, what is c in terms of a and b?

A. a + b - 2
B. a + b + 2
C. (a + b)/2
D. ab - 2
E. ab + 2

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Choice B.

Simplest way in my opinion is to test numbers for the exponents.

We must multiply 37 and 58 long form to confirm that it yields 2146.

If a,b =1, then 2146 = .2146 * 10^c.

Remember when moving decimal to the right, we increase the power of 10 by 1. So c = 4.

A,B = 1, C = 4. The only answer choice that fits this is Choice B.