Bosco

   How many positive integers are there whose digits strictly decrease from left to right, and the sum of the digits is 6?     

   60     51       42       321     

            510     420     3210     

   I thought of these seven . . . maybe you can come up with some more.     

_.     

Adam has 16.45 kg of flour, and he uses 6.4kg to make hot cross buns .The remaining flour is exactly enough to make 15 batches of scones.How much flour,in kg will be in a batch of scones?     

          16.45     original amount of flour     

            6.40     used to make hot cross buns     

          10.05     remaining flour     

          10.05                         enough to make 15 batches of scones     

          10.05 / 15  =  0.67     what it takes to make one batch of scones     

_.     

A bag contains 10 counters. 6 of them are white. a counter is taken at random and not replaced. A second counter is taken out of the bag at random. Calculate the probability that both two counters are white.     

Probability first counter is white                6 / 10     

Probability second counter is white           5 / 9     

Probability both conditions occurring       (6 / 10) • (5 / 9)  =  30 / 90     

                                                                               reduces to 1 / 3    or  33 ¹/₃ %       

_.     

Alex chooses a number at random from the set {1, 2, 3, 4, 5}.  Winnie also chooses a number at random from the same set.  (They can choose the same number.)  What is the probability that the product of their numbers is at least 4?     

The following are the 25 pairs that could occur.     

1 1    1 2    1 3    1 4    1 5     

2 1    2 2    2 3    2 4    2 5     

3 1    3 2    3 3    3 4    3 5     

4 1    4 2    4 3    4 4    4 5     

5 1    5 2    5 3    5 4    5 5     

The highlighted pairs are the 20 that multiply to at least four.     

The probability of choosing a highlighted pair is 20 / 25 which is 80%.     

_.     

The 450 juniors at East HS drink a total of 1500 cartons of milk per 5-day week. If the average senior drinks milk at the same rate as the average junior, what is the average number of total cartons of milk the 600 seniors drink each school day?     

1500 cartons for 5 days works out to 300 cartons for 1 day     

spread among 450 juniors, so each junior drinks ²/₃ of a carton per day.     

The seniors drink milk at the same rate; however, there are 600 of them,     

so the senior class drink ²/₃ x 600  =  400 cartons for 1 day     

_.     

In triangle $ABC,$ $\angle B = 90^\circ.$ Point $X$ is on $\overline{AC}$ such that $\angle BXA = 90^\circ,$ $BC = 15,$ and $CX = 5$. What is $BX$?     

In triangle ABC, angle B = 90^o.  Point X is on AC such that angle BXA = 90^o, BC = 15, and CX = 5. What is BX?     

BXA=90^o therefore BXC=90^o.  BXC is a right triangle with angle BXC = 90^o , CB = 15 , CX = 5.     

Per Pythagoras, c² = a² + b²     ——>>     b²  =    c² –  a²     

                                                                 BX²  =  15² – 5²     

                                                                 BX²  =  225 – 25     

                                                                 BX    =  sqrt(200)     

                                                                 BX    =  14.14     

Does the answer conform to the Triangle Inequality Theorem     

Are CB + BX > CX     15 + 14.14 >   5     

       CB + CX > BX     15 +   5      > 14.14     

       CX + BX > CB       5 + 14.14 > 15     

_.     

What is the corect answer for 10 - 5 2/3? I am stumped BIG time. can you help me?     

      There's a few ways to do it.  If you understand what 2/3 is,      

      I think the following method would probably be the easiest.     

      Look at the 10.  Consider that as 9 ³/₃ and write it down.     

      After you write down 9 ³/₃, then subtract 5 ²/₃ from it.     

                                        9 ³/₃     

                                        5 ²/₃     

                                     –––––––     

                                        4 ¹/₃     

The store is selling lemons at $0.79 each. Each lemon yields about 2 tablespoons of juice. How much will it cost to buy enough lemons to make three 9-inch lemon pies, each requiring half a cup of lemon juice?     

       Per an authoritative conversion reference,     

       one cup of liquid equals 16 tablespoons.     

       A half cup is 8 tbsp, 3 half cups is 24 tbsp.     

       At 2 tbsp per lemon, it'll require 12 lemons.     

       12 • 0.79 $  =  9.48 $ will be the cost.     

_.     

A triangle has integer side lengths 2, 5 and x. What is the area of the triangle if x = 6?     

I think it'd be easier to go with Heron's Formula than trying to manipulate trig functions.     

        s  =  ¹/₂ • (a+b+c)     

        s  =  ¹/₂ • (2 + 5 + 6)  =  6.5     

        A  =  sqrt ( (s)(s–a)(s–b)(s–c) )     

        A  =  sqrt ( 6.5 • 4.5 • 1.5 • 0.5 )     

        A  =  sqrt ( 65.8125 )     

        A  =  8.1125     (as a fraction, 8 ⁹/₈₀ )     

_.     

Find the sum of 1 + 1/2 + 1/5 + 1/10     
The answer should be in the form of a decimal rounded to the nearest HUNDREDTH.      

                     1       =   1.00      

                     1/2    =   0.50     

                     1/5    =   0.20   

                     1/10  =   0.10      

add them up          =   1.80