A. In 1 year Kristen will be four times as old as Danielle. Ten years from then Kristen will only be twice as old as Danielle. How old is Kristen now?

B. The greater of two integers is 4 times the lesser. Of the lesser integer is increased by 12, the result is 3 less than the greater.Find the integers.

C. One number exceeds another by 12, and the sum of the smaller number and twice the larger number is 93. Find the numbers.

Let present age of Kristen be x and of Daniel be y.Then after 1 year,

x+1=4(y+1)or x-4y=3.....Eq.1
After 10 yrs from then, ages will be
(x+1)+10=2[(y+1)+10] Or x+11=2y+22
Or x-2y=11 .....Eq.2. Subtract eq. 1 from eq.2 and solve. We get y=4 and x=19.
Check: After 1 year Kristen will be 20 and Daneil will be 5. After another 10 yrs K will be 30 and D will be 15. OK.

Let smaller and greater integers be x and 4x. Then x+12=4x-3 or -3x=-15
Or x=5, and larger integer will be 20.
Check: 5+12=17=20-3. OK.

Let smaller no. be x, so larger one is x+12. Now x+2(x+12)=93
Or x+2x+24=93 or 3x=93-24=69
Hence x=69/3=23 and larger no. is 23+12=35. Check:23+35*2=23+70=93, OK.

A. Well, I was never good at math, but I can give it a shot! Let's call Kristen's age now "K" and Danielle's age now "D." According to the problem, in one year Kristen will be four times Danielle's age, so we can write the equation K + 1 = 4(D + 1). Then, ten years from now, Kristen will be twice Danielle's age, so we can write the equation K + 10 = 2(D + 10). Now, all we have to do is solve these equations to find Kristen's age now. But since math isn't my strong suit, maybe it'd be easier to just ask Kristen herself? She might have a better idea!

B. Ahh, the classic "greater than" and "lesser than" integer dilemma! Let's call the greater integer "G" and the lesser integer "L." According to the problem, G = 4L. And if we increase L by 12, we get G - 3, so we can write the equation L + 12 = G - 3. Now, let's put on our detective hats and solve this mystery! Or maybe we can just ask Sherlock Holmes for help, he seems pretty good at figuring things out.

C. Oh, the never-ending mysteries of numbers! Let's call the smaller number "S" and the larger number "L." According to the problem, L = S + 12. And if we add twice the larger number to the smaller number, we get 93, so we can write the equation S + 2L = 93. Now, time to put on our detective hats and solve this numerical puzzle! Or maybe we can just call up Nancy Drew, she's great at cracking cases.

A. Let's assume Kristen's current age as K and Danielle's current age as D.

In 1 year, Kristen will be four times as old as Danielle:
K + 1 = 4(D + 1)

Ten years from then, Kristen will only be twice as old as Danielle:
K + 11 = 2(D + 11)

To solve this system of equations, let's simplify them:

Equation 1: K + 1 = 4D + 4
Equation 2: K + 11 = 2D + 22

From Equation 1, we have: K = 4D + 3

Substituting K in Equation 2, we get: 4D + 3 + 11 = 2D + 22

Simplifying further: 4D + 14 = 2D + 22

Combine like terms: 2D = 8

Divide both sides by 2: D = 4

Now, substitute D = 4 into Equation 1 to find K: K = 4(4) + 3 = 19

Therefore, Kristen is currently 19 years old.

B. Let's assume the lesser integer as L and the greater integer as G.

We are given that the greater integer is 4 times the lesser integer:
G = 4L

If the lesser integer is increased by 12, the result is 3 less than the greater integer:
L + 12 = G - 3

Since G = 4L, we can substitute G in the equation above:

L + 12 = 4L - 3

Combine like terms: 3L = 15

Divide both sides by 3: L = 5

Now, substitute L = 5 into G = 4L to find G: G = 4(5) = 20

Therefore, the integers are 5 and 20.

C. Let's assume the smaller number as S and the larger number as L.

We are given that one number exceeds another by 12:
S = L + 12

The sum of the smaller number and twice the larger number is 93:
S + 2L = 93

Substituting S from the first equation into the second equation:
L + 12 + 2L = 93

Combine like terms: 3L + 12 = 93

Subtract 12 from both sides: 3L = 81

Divide both sides by 3: L = 27

Now, substitute L = 27 into S = L + 12 to find S: S = 27 + 12 = 39

Therefore, the numbers are 27 and 39.

A. To solve this problem, let's assign variables to the ages of Kristen and Danielle. Let's say Kristen's current age is K, and Danielle's current age is D.

According to the problem, "In 1 year, Kristen will be four times as old as Danielle." This means that in 1 year, Kristen's age will be K + 1, and Danielle's age will be D + 1. So, we can write the first equation as:

K + 1 = 4(D + 1)

The problem also states that "Ten years from then, Kristen will only be twice as old as Danielle." This means that in 10 years, Kristen's age will be K + 10, and Danielle's age will be D + 10. So, we can write the second equation as:

K + 10 = 2(D + 10)

Now we have a system of equations with two variables (K and D). We can solve this system of equations to find the values of K and D.

Step 1: Solve the first equation for K:
K + 1 = 4D + 4
K = 4D + 3

Step 2: Substitute the value of K from Step 1 into the second equation:
4D + 3 + 10 = 2(D + 10)
4D + 13 = 2D + 20
2D = 7
D = 7/2
D = 3.5

Step 3: Substitute the value of D from Step 2 into the first equation to find K:
K = 4(3.5) + 3
K = 17

Therefore, Kristen is currently 17 years old.

B. Let's say the greater integer is G and the lesser integer is L.

According to the problem, "The greater of two integers is 4 times the lesser." We can write the equation as:
G = 4L

The problem also states, "If the lesser integer is increased by 12, the result is 3 less than the greater." We can write the second equation as:
L + 12 = G - 3

Now we have a system of equations with two variables (G and L). We can solve this system of equations to find the values of G and L.

Step 1: Substitute the value of G from the first equation into the second equation:
L + 12 = 4L - 3

Step 2: Solve for L:
12 + 3 = 4L - L
15 = 3L
L = 5

Step 3: Substitute the value of L into the first equation to find G:
G = 4(5)
G = 20

Therefore, the two integers are 20 and 5.

C. Let's assign variables to the two numbers. Let's say the smaller number is S and the larger number is L.

According to the problem, "One number exceeds another by 12." We can write the equation as:
L = S + 12

The problem also states, "The sum of the smaller number and twice the larger number is 93." We can write the second equation as:
S + 2L = 93

Now we have a system of equations with two variables (S and L). We can solve this system of equations to find the values of S and L.

Step 1: Substitute the value of L from the first equation into the second equation:
S + 2(S + 12) = 93

Step 2: Solve for S:
3S + 24 = 93
3S = 93 - 24
3S = 69
S = 23

Step 3: Substitute the value of S into the first equation to find L:
L = 23 + 12
L = 35

Therefore, the two numbers are 23 and 35.