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question:Bruce purchased 8 kg of grapes at the rate of 70 per kg and 8 kg of mangoes at a certain rate per kg. He paid 1000 to the shopkeeper. What was the rate per kg for the mangoes?
answer:First, let's calculate the total cost of the grapes. Cost of grapes = rate per kg * quantity in kg Cost of grapes = 70 * 8 Cost of grapes = 560 Now, let's find out the total cost of the mangoes. We know that Bruce paid 1000 in total and we have already calculated the cost of grapes. Total cost paid = cost of grapes + cost of mangoes 1000 = 560 + cost of mangoes Cost of mangoes = 1000 - 560 Cost of mangoes = 440 Now we know the cost of 8 kg of mangoes is 440. To find the rate per kg for the mangoes, we divide the total cost by the quantity. Rate per kg for mangoes = cost of mangoes / quantity in kg Rate per kg for mangoes = 440 / 8 Rate per kg for mangoes = 55 So, the rate per kg for the mangoes was boxed{55} .
question:writing the integers from 20 through 199 inclusive, how many times is the digit 7 written?
answer:To solve this problem, we need to count the occurrences of the digit 7 in both the units and tens positions from 20 to 199. 1. **Count in Units Digit**: - The digit 7 appears in the units place in the numbers 27, 37, ..., 197. - These numbers form an arithmetic sequence where the first term is 27 and the common difference is 10. - To find the number of terms, solve ( a_n = 27 + (n-1) times 10 leq 197 ). - Rearranging, we get ( 27 + 10n - 10 leq 197 ), which simplifies to ( 10n leq 180 ). - So, ( n leq 18 ), meaning there are 18 numbers. 2. **Count in Tens Digit**: - The digit 7 appears in the tens place in the numbers 70-79, 170-179. - Each range contributes 10 numbers, so there are 2 ranges, which gives ( 10 times 2 = 20 ). Adding these, the total number of times the digit 7 appears from 20 to 199 is ( 18 + 20 = boxed{38} ).
question:Given the sets M={xmid0 < xleqslant 3} and N={xmid0 < xleqslant 2}, the statement ain M is a ____ condition for ain N. A: Sufficient but not necessary condition B: Necessary but not sufficient condition C: Necessary and sufficient condition D: Neither necessary nor sufficient conditio
answer:We are given the sets M={xmid0 < xleqslant 3} and N={xmid0 < xleqslant 2}. Notice that Msupseteq N. 1. If "ain M", we cannot conclude that "ain N". For example, a=2.5in M, but a notin N. 2. However, if "ain N", then we can conclude that "ain M", as every element of N is also an element of M. Based on the above analysis, the statement "ain M" is a necessary but not sufficient condition for "ain N". Therefore, the correct answer is boxed{text{B: Necessary but not sufficient condition}}
question:There were 28 bales of hay in the barn. Tim stacked more bales in the barn today. After stacking, there are now a certain number of bales in the barn. Tim stored 26 bales in the barn. How many bales of hay are in the barn now?
answer:If there were originally 28 bales of hay in the barn and Tim added 26 more bales, then the total number of bales in the barn now is: 28 (original bales) + 26 (added bales) = 54 bales So, there are now boxed{54} bales of hay in the barn.