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Yes Jesus answers prayers, it is through his grace, that we receive what we need. As my pastor Dollar puts it, when were in the old Mosaic covenant everything was about our works (for instance! we had to give burnt offerings and sacrifices. Now that Jesus came with his Grace now it is not by our works that our prayers are answered but by our faith and us allowing God to do what needs to do. We have to allow The Lord to do his will with us.
When we pray for something to God, our last sentence should always be "But Lord let your will be done and lead me as you please "
If this thing we keep in our minds I think all our prayers will be answered in the way God, our creator, loving father intends us to live. This way may be not exactly the way we intended God to work in us. But then God is the ultimate supreme being who knows what is best for us. We must surrender to him as Jesus our Lord surrendered himself even in suffering,
humiliation, torture and in the most unjust death he got. This is tough to believe. The suffering may go for years and we may wonder whether God listens to us, whether he is really there or why all our efforts go in vain inspite of our every attempt to succeed or make it happen.
In my case I have witnessed quite many instances where God has answered my prayers right from the age five. It may look ridiculous or even like faking, but there is no shame in sharing what God has done to me!
When I was just five, in our place (Mangalore in Karnataka, India) there was a custom that if anybody is a victim of evil eye then we must pray to God clutching a pinch of salt and couple of dry chillies, rub it over the body of one who is affected and then throw it in the fire. The victim was back to his health as the salt and chilly burnt in the fire and if no smell of burning chillies came. As a child I used to see this my mother performing it on my body and even with other babies and we all used to get well. The wonder was that, there was no any smell of chilly even when they were burning in the fire!
As a child I couldn't interpret it then.
But the ritual caught my eye and I experimented it on a Jack fruit tree.
There was a tall Jack fruit tree standing in front of our house and its fruits on ripening used to rotten, become black and fall down. We had several other Jack fruit trees but none were like this. I asked my mother one day, what was the reason for the fruits to fall like that. She told me it used give the best fruit among all the trees and one day, one carpenter after eating the fruit said, "This is the sweetest fruit I ever ate, I will take some of them next time". As soon as he left the house, the fruits from the tree started falling one after another. Since then every year we have this problem with this tree and the fruits get rotten and fall as soon as they become big" I asked her when that happened. She said, " Five years back".
So, when nobody was there at home, I took some salt and 2 chillies and a cross in my hand, rubbed it round the Jack fruit which was almost touching the ground, recited some prayers mother had taught me, Our father and and hail Mary and threw the chillies and the salt in the fire. That year not even one Jack fruit got rotten and we almost had over 50 fruits that year from that tree. While eating my father said " It is almost after 5 years we are eating such sweet fruit". then I said to them what I had done and how I prayed Our father and Hail Mary to make it alright. It was a miracle and the way God answered a child's prayer. Then I was taken around in the neighborhood to heal the coconut plants, Papaya trees which were affected by evil eye and I almost healed all of them barring a very few which were not yielding because of not watering them at all.
We have heard God listens to the prayers coming from an innocent and pure heart. Well, after that I never been able to perform any such things as I grew up!
So I can conclude that if our hearts are pure and holy God will certainly listen to our prayers. If God perceives something is good and necessary for us he will surely provide us. But he may give it to us not when we want it but when he thinks it best to give us.