| Exercise-XIX |
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| 2. |
If (a, b) and (a + 3 , b + m) are two points on the line defined by the equation x = 3y – 7 then m = |
| Sol: |
x |
= |
3y – 7 |
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put x = a , y = b , x = a + 3 , y = b + m |
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a |
= |
3b – 7 |
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a – 3b |
= |
– 7 |
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x = a + 3 , y = b + 4 |
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a + 3 |
= |
3(b + m) – 7 |
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a + 3 |
= |
3b + 3m – 7 |
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a – 3b |
= |
3m – 10 (bu t a – 3b = – 7) |
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– 7 |
= |
3m – 10 |
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3m |
= |
10 – 7 |
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3m |
= |
3 |
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m |
= |
1 |
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| 3. |
If David loses 8 pounds, he will weigh twice as much as his sister. Together they now weigh 198 pounds. What is David's present weight in pounds ? |
| Sol: |
Let David's weight be x pounds |
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David's present weight |
= |
x – 8 pounds |
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sister weight |
= |
 (x – 8) pounds |
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x – 8 + (x – 8) |
= |
198 |
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2(x – 8) + 1(x – 8) |
= |
198 × 2 |
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2x – 16 + x – 8 |
= |
396 |
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3x |
= |
420 |
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x |
= |
140 |
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Present weight of David |
= |
(140 – 8) |
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|
= |
132 pounds |
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| 4. |
The cost of 12 maps and 10 books is $38 and the cost of 20 maps and 15 books is $ 60. What is the difference between the prices of each item ? |
| Sol: |
Let cost of a map be $ x |
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Let cost of a book be $ y |
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 |
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12x + 10y |
= |
38 |
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12 × 1.50 + 10y |
= |
38 |
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18 + 10y |
= |
38 |
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10y |
= |
20 |
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y |
= |
2 |
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y – x |
= |
$ 2 – $ 1.50 |
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|
= |
$ 0.50 |
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| 5. |
A supermarket's total sales for the month of February of this year was $ 384. If the total sales for the same month last year was $ 320 million. What was the percent increase in sales ? |
| Sol: |
Increase in sales |
= |
$ 384 – $ 320 |
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|
= |
$ 64 |
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% increase |
= |
 |
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|
= |
20 % |
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| 6. |
List I : 3 , 6 , 8 , 19
List II : x , 3 , 6 , 8 , 19
If the median of the List I is equal to the median of the numbers in List II , what is the value of x ? |
| Sol: |
The median of List I is  = 7 |
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The median of List II is also 7 |
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The median of x , 3 , 6 , 8 , 19 is 7 |
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As the median of list II is 7 |
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3 , 6 , x , 8 , 19 are the numbers |
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x = 7 |
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| 7. |
The present ratio of students to teachers in a school is 30. If 50 new students and 5 new teachers join the ratio of students to teachers would then be 25 : 1. What is the present number of students? |
| Sol: |
Let the number of students : teachers |
= |
30 x : 1x |
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30x + 50 : x + 5 |
= |
25 : 1 |
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(30x + 50) |
= |
25(x + 5) |
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30x + 50 |
= |
25x + 125 |
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5x |
= |
75 |
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x |
= |
15 |
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Number of students |
= |
30 × 15 + 50 |
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|
= |
500 |
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| 8. |
What is the smallest integer n for which 25n > 512 |
| Sol: |
(52)n > 512 |
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(52)6  512 |
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(52)7 > 512 |
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514 > 512 |
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smallest integer n = 7 |
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| 9. |
If 784 = 2a × 7b then the value of 5a + 6b = |
| Sol: |
784 |
= |
28 × 28 |
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|
= |
7 × 4 × 7 × 4 |
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|
= |
72 × 42 |
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2a × 7b |
= |
72 × 24 |
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a |
= |
4 |
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b |
= |
2 |
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5a + 6b |
= |
5 × 4 + 6 × 2 |
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|
= |
32 |
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| 10. |
If p is the product of the integers from 1 to 20, inclusive what is the greatest integer k for which 5k is a factor of p ? |
| Sol: |
1 × 2 × 3 × 4 × 5 × 6 × 7 × 8 × 9 × 10 × 11 × 12 × 13 × 14 × 15 × 16 × 17 × 18 × 19 × 20 |
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Product of multiples of 3 |
= |
3 × 6 × 9 × 12 × 15 × 18 |
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= |
3 × 3 × 2 × 3 × 3 × 3 × 4 × 3 × 5 × 3 × 3 × 2 |
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|
= |
38 × 2 × 4 ×5 × 2 |
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Greatest integer |
= |
38 |
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k |
= |
8 |
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SAT Math 
